Mathematical Models of Electric Machines

I. P. Kopylov

Mathematical Models of Electric Machines

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Mathematical Modds of E1edric Machines

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I. P.Kopylov

Mathematical Models of Electric Machines Translat ed f ro m the Russ ian

by P . S . IVAN OV

M ir Pub lishe rs M os co w

F ir st tllt1Jl;5h~d 19M.

n e vised [rom lh ll 1980 Russian l'diUOll

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Tran s literation

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X : mll ch in ~ with co nsi de r ati on for th e effec ts of s aliency, Iliscr elene.ss of th e windi ng s tructu re, s al u ra t ion, an d in d uce d! currsn Ul, The equaucue Cu r all loo ps in rho per mean ce meth od repr esenreuon do not neceesnatc IIdd itiplInl coor di na te tra nsfor mations, Alt hough t he pl,.,gress in th l.' de volo pmout of etee rrto mach i ne models on th e b llsi ~ of rieh! eq ua tfons jlZ a pp reel ahle, t he meet meteria l ad vancements life mad e by use of the equ ation!' wr itten i n uie notati on adop ted i n elce t rtc circuit &hOOl)" T her efore in t he fu rthe rpreeeuta u c n of th e lcxt we will bas ic all )' e m ploy the eq uations of tbe ge nera lizocl etcc t rc meehanica l energ y ccnve etee.

1.4. The Primitive Fo ur ..W in dJng Machine All electric mac hin l.'S ere identic al in t ho 5ef15ll t hat th oy ecnvere, eOf:'rgy from elecret eel to medulII ica.\ fo rm or Ieom Ull'(;halliul to cleet rietll form . Bu t e loctr ic mar hili es even or the sa me se ries d iHe r from one anothe r in perfor m ance. Th e hosie t y pes of eteer elc mach i nes c>.n be red uced to II ge nerahaed , or prfml t.tve , me del re presenti ng :I set of two pairs of wi nd ings. moving with respec t t o eac h ethe r, I n Fi~ , 1.11 is shown t he ide ali zed model of a sy mme tri c, mach in o 118\'i ng II smoo th nir-g llp s rruc l ure lind s inusoid III wind ings . wi th. -t h'e perm ea nce eq ual t o zero. A sinusoid llll y varying ve ltuge a pplied to th e winding prod ucesa etrcu te e fio l d in t he air gap. W ith t he wind ings being sym met r ic. I ~i n Wl o i d n l s y mmet ric vo l Lllge sets up II ~ i nusoidal fi eld in th e ge p. Th e term ' pri miti ve machtne ' sta nds for an Id~aliKd lu;o-pol~ ttro-fAQ$~ I/lmmei rlc (bal anttd) machLne haVing on, pair of /rim/ings Dn th~ TOtor and th~ olher fXJir on Ou: stator l'S s hown in F ig. 1. 21. Here wU' w~ a re t he U nto r wind ings. el ollg t he cz and ~ axes ; w;. ~ are t he rotor wind inas 810ng t he IX a Dd exes; uj, u: . are voltag es al ong the cz ea d ~ u:l.'li on th e ! t a to r and ro tor reepeeth'ely; end CIl. is t he a ngu l8r s peed or t he roto r. Th e Ilnillysis of th e two- pole machine M a model en1Ulles us to exte nd the fl.'S ults lind desui bc t he ' pr ocesses or.curri ng in a real multi pol ar machin e. T he t wo-p hase' marhine bes {ou r wi ndi ngs

r-

u:..

u,

.and is descri bable by four vol t age equa t ions (8 mi nimu m nu mber -of eq ulltions in eompeeisen wi Lh t hos8 used for describing Si llj'le-phese, t hree-phase. an d m--p hase machi nes). Conside r an idealtzed 'lIoiCorm-lIir ifap machi ne ' whose wind ings 11I"O t aken lo be in the 'form of cu rrenl ' hools ":bera t he mod distri but ion Is s lnU50idal. 'Our ide., lind m ncblne hllS:nO sa tu ration, no r non lin ear resetances, .and t herefore el:h lbits a ~i nusoid al field in t he alr gllp ....hen t he w ind ings ore fed wi t h sinuso idal voltage. T he Ideali zed ma.cbin e mooel is t he analog 01 an i nd ucti on mac hi ne when t hestah.r windi ngs", and Ultl aceept sin usoidal Yollaaes

,

:nt frequenc y f" 00" a part in t hn D. T he rotor wi ndiogt ca rry euerenta produced b y th e voltage ap plied to t he rotor or indueed by t ho e:urre nts in the stator win d ings. In an Ind uetion machine, tho roto r ang ular spelld is 11)• .p 11). (w . is t he synchro;II OIU speed of t he fiel d), t nd the rctce and stator fiel ds are st..atio narr. "Wi th respeet to eac h oLhh sinc e Lhe mech a nical roto r speed 11). plu sl mlllU5 t he rotor field sP,oed relative to (0), is equal to til • • T lte idea lized machin e model filpres&nls a sYllchronoU5 machinl if an ee voltage Is put 'Cl"Oi'lS the s retcr windi ngs and a dt. voltag, .aCf0S5 the rotor w ind ints, and vice vcrsa. Here (0) . = ClJ. , t.e. t he :stalor lind ro tor ftelds '8l'8 stational')' with mpeel to eac h other. If II de voh ai e d rives eqrfilnl t hrough the s tator windi ngs. t he rolor fi el d t ravels in t he dirK t ion opposlte to t hat of t he rotor, 11(1 the ~tato r an d rot or field! are s tatio nary relative to t be sla t ionar y refer. enee fra mo. With de ~u p pl y l o all t he wind ings, it is e nough t o

-01 fr equency I, = I,., 8il her

sa

l A. Th. PrimitivD Foue-Wil1c1iI19 M . " hlnD

ha ve one field win di ng i ll which t he; ms ul tanl. m agnetizing force 13 equd to t be geo metric s um of th e magne thiog forces of each winding. 11l de mac.hines, t be arma ture winding ca nies a mu!ti pb l3e eneeIll ling curren t reetili&cl meehll.nicall ~ h y means of a commutaLor • frequency ee c verter (Fe). By red uci PI: a pol y phllSe s}'Stem to a rwoph ese 006. we o bt ain l.he model of iii de mach ine (FIg. 1.12). As in II synchronous machine, t he a rma tu re fi eld of t he de mac hi ne rotates ) «

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Fe

Pia. 1. 12. Th e mooe l of. de IJIxhlM Ilod an 8C colltmn l.alor macbiof

in t he op posite s ense wi t h respec t to th e ar ma t ure. ' Vhen CIlr ... fit. the ar mat ure field is lIta tio no ry rela five to t he fiel d wlndi n: a nd to th e !ll ation nry referen ce Ir nme. I t. llllouid be note d t hat t ho sl i p In s ync hrcnoua mac hines and de ma cbfnes equ als taro . A com mutator can be repl aced by a aumi conductoe Ieeq uency convertor, re ed rola y conv erter, etc. T he processes of energ y conversion in t he air go p do not cha nge wit h t he repla ceme nt (If one t ype of Fe by t ho o the r. However, a co nventi onal com mutator holds a fixed ti e bet wee n t he frequ ency a nd t he ro tor speed (o)r• .wh ile a se mtcond ucre e Fe may afford the possi.bi li ~)' of eecurtng contro ll abl e (eedback rc regu ll'lte I; aecording to CIlr. A!I regards its power s u pply, a semicond uctorcommut ator mach i ne i.! Il. de mac hine. Historically. this l.yP'l of de ) _, I li a

"machine received several, names-c-eectif tee-rype mechtn e, se mtecedl,u::tor-.commutl tor machine. ee nreeness mac hi ne, etc. I n an lie comm ut ato r machine, al te rn at iog curre nts 61i.s t in t he st ator a nd rotor .....indl ngs. a Dd th e frequ enc Y converter t rallsfor ms t b o al te rn _Li ng eureem at lh . h us frequ ency in t o l h_ t of sl i p Ireq ueney (see Fig. 1. 12). .A! in other electric. machi nes, bere th o sta tor field iS lltAtio nary re lattv e t o t he ro tor fiel d . These machines.. een bfl of t he l ingle--phaso, th ree- phase., or mul tt pbese t )' pe!; th e s t at or lind ro tor windin p can be connected in series or paraUel . or ean have magne\ic coup ling. TIle primitive machine with a rotor s peed (I), - 0 can rep resent a n electeome gneuc eou veuer-c-e tra nsform er. I n t his COlle i t is s ufficient to cons ider se paratel y th e pair of wiudlnga on t he sta tor ond rot or olong the a axis or '" axis because wit h tb e roto r a t stands t tl! t here is no cou pli ng bet ween t he windings llhih ed 9O~ llpart in s pace, Although t ransformers perform electr omagnet.io ec nvaeslon of energy. t he )' belong to electric machines because of th e gene rali t}' of eq uations and for historical reasons. Tile classifica ti on of oloct ric mee hiu es hlto i ndi vidual types i.!J lare e.l y conventi onal. O ~e and t he s ame machine can cpe eete M e s ynchrcnous and lUi an ¥ ynclu'oDous mac hine. In etecter c mac hines t here occurs elC!(;tro mt'Chan ical and eledromaane tic e-ue!"ln'ec nverslc n simultaneously. Tho processes of elec1romechlUlicai e nerg y conversio n in the prjmilivo ma chine are described by vol tage equatioos (1.34) and equatioll 01 motion (1.35) ,.~ + (dldt )

(d ld t) M

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(d /dt )J11

0

0

,.:' +(dldt ) L~

L,fIl ,

M (fJ.

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,.' + (d ldt )LD

(dld t) iU

o

(dl d t ) M

o

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+ (dldt) L~

l~

I~

( 1.34)

(lIp ) ,J d fll , ldt ± M, - Jot .

(i .35)

Bqs. (1.34) and (1.35) t9gethor wit h t he equa ti on lo r an elect romagnetic torque lorm the ifundll. ment ll1 system of e-qu ations of elec tromecho nical energy ccnve rslen. In Eqll. (f .M ). uj-. u:.. loll . t:.. 4 . ':.. I~ are th e " oltages a nd curr ents in t he sta to~ a nd rotor Vo'indl op o n th e 0: and II axes respecti vely ; ~ . ,.j . "0. Ii lire th e re:sistlnces of s tator and rotor win dinp rospecLh'ely; mutual inductance ; and ~ . LA. L:.. Li I re total i nd uctances of t,he l tll.tor an d rot or 'W Indi ngs a lool t he a. and tJ Uet respectively.

u.:..

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35

Wi ndi ng in ductances ar e d efi ned by t ile kno wn relati on s L".1

- M+ "...

L/ - M +',.

;" = M + I'..

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t, ~ M+'!

('-36)

whOlO 1:.. 1~. [~ . 1~ a NI leakage iDd~ctances DC the sta to r and rot or wind ings alon g th o a and p ax es ees pee uvely. Th e mu tu al i ndu ct a nce lind le ak age inductance.. are Cou nd by th o known me t hod s In vol ving tho (' a1 (';U~lI. ti o n s or ex per imen tal nnal ys is, r.e. using eq uiv a leu t circ uits a nd desi g n formu las. T ho ass u m pucu is t hat t here is (I w orkin g tlu x 'i'h iclL li n ks t he s t a t or and rotor wind i ngs and e lso lea kage fl u xes li n ki ng o nly OM wind ing. E quation s (1.34) desc ribe a hy pp t be tica l ma ch in e h ay ln!: t he !l3me n umber of luTJUI on the stator ll.p d o n t he rot or, with th o windings beiog pse u d ostaU ona ry. T o pres er ve t h e po wer In v aeiance in 1111 ac l u al m achi ne and i n t he IIH\r vici ng a few peri pheral uo tts, ehe nnel-tc-che n uel ad a ptersprovi d ing d ir ect con ne ction betweenthe selec t or cha nnels of pr ocessors, a nd multipl exors for da ta trll ll~fe r over a fow chan nels. Per tphe ral units inc lude magnet ic-tape , ma gnetic-d r um. or d tode st orages; in pu t -ou t pu t. dev ices for Pillich cards an d ta pes ; pri nters ; dal,n terminats and consoles; tbe means [mo dems} for data t eleprocessi ng a nd commun ication witll ,cont rol units . The mut hemn tfc a! program pac kage (soft ware) for th e EC com puter includes t he programs of t hree ce tegceres : opcraucg sys tems ensuejug t he Hnk between t hll o pera tor audmser a nd di s ~ri b u tillg the jobs. lin d system resources: ma intona nce programs or lest rout ines (de b ugging. checklng , a nd d iag nol:ti c rou t ines): nut! t ho packs of a ppttcnt lon ,

prograJrul , w hich a rc fu net ion :lll y co mp le te se UI a r ra nged for t l18 so lutio n o f a dofi nite cla ss 'of prob lems. T he EC u n ita r y sys te m te pres oo ts a fa m ily of progr am-co mpati ble comput in g se tcpa of the foll owi ng I)"pea : EC- t OID. EC·l020,

EC-l02 t ,

EC-t 03O, EC-tQ410. EC-I05O . EC-IOOO. The workin g

s t orage capacit y ranees from 8 t o 10' ki ]ob)' les. T h e b llBlc featu re s 01 the EC ce m puter aro it s uni versal ity . eda pl abili t y for V::lriow a ppJica l io ns. an d th e possib ility of II. grA d ua l bu il d u p of tho com puting poWer over II. wi do fan go. Th e versat il i t y Is d uo to t ho in struction se t i nvo l v ing fi xed-po in t and lJo!lling. po in t ce m pu tet tons . logic And d ccl ma l ope ra t iOIl!!, ope ra tio ns w it h varia ble-length words, a nd e teo due t o vario us d ol ll Ierm m s , multiprogra mm ing possihilities . and the ad vanced sys tem of soft ware . T he ado pta bil ity for use! "ste m!! from the chnngeahle s t r uctu re of the EC eyete m (roplneeabiJity of mem ori es , ehnn uels. pe riphera l e quipment). A gradUAl Incre ase in tli e comp uting power ce ll be a chie ved by seve ra l method, . name ly, by i ncreasing t he number 01 peripheta l UDit.'l a nd t he work ing s torage cApac it y, produ c ing multim achine cc m p uting com plexes, replacing tbe processor by 1\ futer.s peed t ype, etc. The prOgn\m co mpa t ib il it y of th e EC co m r uter co m ~s f rom the unUled 10giCflI s tructu re (s ta oda rd il.a t io n 0 t he in st n lclle n se t, da ta re presen ta t ion form , And add ress system) . D urini the las t 25 ,.~a rs ;t ho com puter speed. s to rage ca pac ity, and reliAbili ty h a ve i ncre ased m llDY ti mes . Th e o1:u a ll di mens ions , th o e nerg )' co nsumed , a nd the specific cost of co mp ute rs decrell!M! very fAst eoncun-enl wil h th r improvem e nt of t he ir pa tllmeterS an d c haracteri s tics. At Lhe ato r t of the 19705, the firs t fourt h-I en.u lHio1/. computers appea red , wht eh began to \l.se medium-scale ICs (a bout 100 gates o n a chip) a nd large-scale rqs (t housands of gates o n a ch ip). Wh at ,d iat inl:ulshos t he Iour t h-genem t .icn computers is t ha t th ay widel y -e mploy se miconductor storages, en lorged Iust.ruct lon se ts , mi cr opro grommi ng , bu il t- in s ubrout ines , au tomat ed program de buggin g, peri phera l uni t e a nd cho nnols of d ivorsed t ype s a nd im proved qu ailt y, in tl"rfoces , specia li>torl .processora . T hose computers exh ibit .e nhe need reliabili ty a nd [oFm t he bas is lor the cc cstrucuon o[ mul Umachi no a nd multiprocessor comp uli ng complexes. The em erge nce of an au to ma ti c uni versa l d lgi tllli com p uter t ha t perform s Ilr it hmetic a nd logical o pera t io ns with a hi gb s peed opens up new q uali ta t ive po!ISibil it ias l or cond uct in: t he t heo reti cal i nvosti ga ti ons in co njunctio n with ebeck ex per iments. Hybri d co mpu ters which Itomprise d ig ital mach ines a nd a nalog -ue vtees hold mu ch promlso for the e ffic ien t combina t ion of the 010me e ts of & h ybrid iD.!lt.Ualio D to enable t he mcst ratiooa l solution -cr prob lems. TIle d igital computer in a hy brid complex ill a co ntro l machine

l.S" Ap p liu tio n 0 1 Co mf>\l'."

wbich s imu lta ne ously gives t he ecurce i Mor m1lt ion for t he Iur thar M1l utioa of pro blema on a nalog dev ices. T h ls complex , when een neeled to t he 5}'.!1le m of ap propr-iate lrl\nstlueorl, ca n contro l an expeellllent a nd keep t he li nk from t he mdme nt of dll. a oalY.!lis 1.0 t ho moment of obtlinint t ho resu lt. T h lS 1150 offor1l Del!.' poS!ibili ties of the search for an c pt lmnl mode of 0~ r8 tion on the pr inci ple of .self- ins t ruction of t he .!IYlIt em. At present the ways are sough t for ihe COll$t ruel iOll of h igll-performloce com puting S)'"Sle ms by us ing , ca&CArled se tu p ~ ... nposed of l llrge nu mber of iden tical u niverse] digl hll computers prog rflmOrlZllQi l l'd for t he n:!a Hntio n of a spet ified al gorithm. The devele pmen t of such s tr uct ures Involves t he ljC-finement of tile reqlli re lJlf' nlS for \'el'Slllilil)', pertoema ueo. t.omput&lioll IlU Ural )'. a lld d iroctl y depends on t he etess of pro blem s t o be so lved . B )' th e ir slruel m'lil, t he fourt h-goner. t ion mac h ines are mu lti proeeaoe se t ups de voted t o t he ro mman mem ory blo ck and t he common ex te nt of peri pherlll devtce e. An to.ggregal e of compu t ioi facillt ie3 forms a cen te r connec te d to nu merou s s ubscet bers by commumeauon linea. Such II. Ile ~wo rk orfeI1J t Ile pDMlb ili t y Jor th e communal U50 of compu ters 1.lr lin indiv id ua l or " group of reseerehers who mill' contac t th e coote r by tele graph or tel ephone from lilly region of the countr y, re la y the message Ior t he solu tton of n pro blem and recei ve the anawer on tl le giv e n do te . Both llllOloll and d igital compu loq fi nd use for t he sot uuo n of problems in ele ctro mechanics . If soma ten eq uat io ns describing t ho transienl processes In electric machtues lire eno ugh an d the par ameters ent eri ng lute t he equa t ions {\~ const llnt , it is well to solve the pro blem s 0 11 lin IInalog compu ter. Wh ore tho numbe r of equ ations i~ IlIqer _ t ile pa ra me ters ar e nonli nM l"lInd t hott) is a need for solVi ng problems for th e optimiza t ion of lin el)erg }" eon ver ter, it is nece.s.!lU'y 10 enocse tI dlll'itni cc mputer , 1/1 1M /1l1l1ly,u oj electric 'rlQChIM', It is tZ pedUflt t o tm pWII both 1If1l1llllit'G1 nuthods and arnllog and d.lgttlll computu,. The tzptr tence in chOOflng Iht combinot ton of mt thbds of anolyfi, determ Ines th, tkgru 01 w:curM!I and prolourulnu, oj the .tOlwloA of the probum. In soh -ing II problem in e leetromeplillnics, tbe ~areher should prima r il y formu lllIc t he equatio ns for t Ilt prceesses under s tu d)' \0 • $Offieien t degree of acc.llr.lcy flIld t llen choose a eem pute e to form I mat hematical med ol . :'Ie xt he !Ihould refine th e medel wt th the a im 10 estimlto t bi! time it woul d ta ke ;to 5OIv8 t he pro blem nnd t he ClXpef.le d .ccuracy of t he solution. T he fi nltl step involves draWing tbe plan of t he ex per inlents \.0 be r UD. Despi te t he ir great oppo rtu nities, .ee mputer facilities can soh-e a rath er Iimite.1 r l ngo of pro blom.!l in eloetro meeha nic.s. T aki ng into accou nt. eve n two or t hreo hllrmon ics i o t he ai r gllp lind lWO or tlu'e6 1oops 011 t he- s tate r and ro to r neeeeenet ee so lv ing a few tens or

..

Ch. 2. EI_d'OI'I'Wld'la nlo;ll1 Energy COnYllnion

equat ioD' . Conseque ntly , th o researcher s ho ul d thoroughl y choose the mathema tical mo de l, keeping in mi nd the pOWeT of co m pu te r teen u see. an d estimate the tim e re q uired for t he solutio n of th e problem a nd the poss ib le solutio n ace u roc)'.

2

Chapte r

Electromechanic;al Energy C onversion Involv ing h' C ircula r Field

, ,,

2.1. The Equations of the Generali zed Ele ctric Machine Coca tde r a t wo-phase t wJ.pole e lectric machine (F ig . 2 .1). It. has t wo orthogo nal sy llte ms of s ta te r a nd rol or windings uf.. w1

"

e, Fir.

z.t.

T he m.chia. mock-I

and an d

ID~ .

w,.

II ..

'rnll ~

In. o on U"fU)a!ormed eoordiJu te ay!tem

raspec U\'el)' , lyiz>1: on ui e ata te r a od Totoor a xel 0 •. b. b, . T he rec llm gula r ' coordi na te rrllimu of t be s ta tor a nd

2.1. The EqueHont of lhe Gene,allzed Eleel'l( Meehlne

45

rotor move with respect to .each other', and t be angle 0 between th e

exes determines th e rela ti ve ro tational vel ocity. Wit h t he st ato r being st atio nary, (2.1) til , = d af~t The d ifferential equati ons of voltages in na tura l or phase (nont ransformed) coord ina tes have t he form

= i ~r~ + d'l'~ {dt ut = itrt + dlfVdt u~

- u:= i :r~ +dlJ': ldt - u~ =

(2.2)

i;;r;; + dr~ /dt

In Eqs. (2.2) , t ile fre qu encies of cur rs uta in t he s tator and ro t or are differe nt , and the 'minus' signs before t he rotor volt ngea denote that tho active power fl ows from th e~ sta to r t o t he shaft (moto r ing act ioo). The flux li nkages of th e wind ings are

'1'; = L; i~ +1H cos Oi: + M si n ei~ lJft .... L~i~

+ Iff coo M -

M s in OJ:

lJf: = L~l:

+ M cos Oi~ -

M s in M

(2.3)

lJf~ = Lb l~ + M cos etj + M s in el~ Hero t he coeffi cients ahea d .of the; currents vary with the same rate as the curre nts . If we substi t u te express tc ns (2.3) (u to (2.2), t he resultant equalions will be too awkw ard and conta in pe ri odic coefficie nts . To slmpltf y t ho so lut ion of equatio ns, it is necessary t o hav e t he sa me Irequeoctes i n the sta tor a nd rotor win dings a nd ensu re tile Invarteoce of power, t.e. to ena ble the pewee coming to t he shaft, the losses , an d the energy consumed t o be. the same as the y are in a rea l mach ine . Look i nt o the pr ocesse s of energy cbn vers ton in a machin e within the ai r ga p (t he s pacing between the rotor and sLato r) which concentr a tes the energy of a magne t ic fiel d. A r ot a li ng field is set up in t he a ir ga p of a real mac hi ne owing to a definite d istribution of the wind ings i n space. a nd to th o t i l!J.e sh ift betwee n cur re nts end vol tages . Wit h t he sin usoidall y vo~y [ n g voltag es impressed on ure ter-minals of a n ide al machine , n l;i~cu lar field app ear s in th e a ir ga p. B y" t ho t hird law of e lectr umechanics, there is a rig id li nk between t he freque ncies of curre nts ' i n the stator and r oto r , t he sta to r end rotor fields heing stationary wit h respect to each other . Accou nt m ust etsc btl t ak en here ofrt he mechan ical speeds of the rot or a nd st a l or. For a st a t ionary s ta to r, IJ) . = 1O, ± lOJ r (where

fiJ,.

" is t he speed of t he rot or fie ld re la t ive to t he rotor) . II is cc n vcm e ut C/>. 2. Eledrom ec han lca l Ene rg y Conve rs io n

to represent t he circular Held, fn the ai r /lap by t he resu lta nt magnetie flux dens ity vecto r Ii' = B~ j B~, ff _ BOo jB~ (2 .4) and by t he result ant nux li nkage vector

+

+

+

+

'Y' "'" 11'i.' ill"6' 0/'" = "Yo. p¥~ (2 .5) T he state r end rotor vo lt ~ g~ s and cu rr ents can bo represe nted es uni t resultn nt vec tors V' , fJ' a nd 7 ' , 1' . respec t i vely . Si nce t he w ind ings in t he pr im itive ma ch ine li e s pace, Lhe vo l tages U~

= 0", si n

lVi ,

ut

= U m COS wI

9lr

apArt i n (2 .6)

un pressed acr oss th e windin gs produ ce tho! res ul ta n t field 8' a nd IV' in t he Hi t gnp. For voltages (2.2) given in terms 01 t he resultant vect ors, t ho equn t tons assum e t he for m ,

+ d'fi' !dl ~ l ' RT + d'f' ldt

U· .., .T'R'

- D'

(2 .7)

H ere jl' = r:. = rjs, Ii' '"'" r~ = r~, Exn mine till' prooeseca of CIJ('I'gy con vcrst ou as viewed from t he cccrdmete reference fra mo rot.a t.ing a l an nrbi tr or)' speed w , (t he observer's s peed), For litis coorA lli ulIle s yste m the power invarlence depends on w< a nd Ire quc uey f •. I ll us trat e ;l change i n frequeuc y with tile rota ti on of l ite cc crd tnatc sy sturn by a n ex am ple of t he commu tator much ine shown >

..,

i n F ig , 2. 2.

III tbi s machine, t he s ta tor cueetee a thre e-phase Wi nd ing 1 , each p h ase w ind ing A , B, and C being fed wi t h eo voltage which prc duces in th e gop a field revofvln g at II sy nch rono us s peed 00 , . T ile roto r c 2 r uns at II s peed w ~ and the frequen c y i n the ro tor win d ing is Fig. 2.2. A commutator macblne I , = 1\$· Th e br ushes rigidly COIIwtt b Nvol vlllg bru shes . necte d to th e coord ina te axes sli do ove r the ro tor wind ing and rotate togeth er with t he br ush ri ng -3 at a s peed 00 , . Th o n umber of br us hes is eq ual to t he n umb er of phas es. As seen from t he figure, the ma-

2.1 , rh . Equelio "s oIth. G.n.reliz&d EI.d rlc Medlin..

up jO.)

{f~ exp 1 (0. - OJ """ R ' I ' exp j (0. _ fe) Takin g the deriva t ives in (2.9) y ields

+ (dldl ) (If'- exp j

+ do/'Idt + I.wj fl> R'I' + d\j!' ldt + j (w. -

fr ... R']'

(2.9)

(0. - O) (2.10)

D' = w.) 'P T ho ob ta in ed vol t ege equati ons for th e res ult ant ve ct or s arcdeHned with res pect to th e coord in ate refe rence fra me ro t atiug at an arbitrary speed wi t h t he rotor. These aro the simplest a nd most genera l Ki rcl lhorrs oqu e ttous Ior th s pr im iti ve m ach in e . T he equati ons ex p resse d i n t he vuet ab le-speed rotat ing re feren ce Iea me arein rare use . Of rno" t iJlt 9nult ure the eq uations wru to n i n coord inate& « au d ~ whe n w. = 0 and t he eq ua tl olls in t ho system of coord inates d and q when w . = w,. T he la t er aya t em is the rotating reference Irnme, wh ich is mos t po pul ar for th e anal ys is ot eynohronons. machines when w . _ W r - w, . The ro t or and s tato r fields nppe ur st at iona ry for t he obse rve r when hOI v iews t hem from th e r ot or. Hero the mod eli ng of e nerg y COli version pr ocesses ln vnl v es di rec t curren ts. For th e s t atio na ry refer ence fr-ame (/I). = 0) , with tho r eferen ce axes ex and ~ be in g rig idl y connected 't o t he st at or , E qa. (2.10) ta keon t h o for m fl o ... R')' + dW' /di (2.11) Dr _ RrTr + awrl d_t _ jw r'¥' r

Reso lv ing the res ult a nt vec tors alon g t ho axes ex an d ~ g lv C!s the equ ations of voltnges for t he primi ti ve m ach ine i n terms of fl ux. linkages: u~ = t~ ,.~ + d1f:., dt

u&= I&rB+ dlflptat ~ = I~ r~ +dW:Jdr

+ tiJ. '1'&

u;' = i ~ '8 + a'VDJd~ -

w. III;;'

(2. t2)

B y !!ubs Ut uti ng t he fJU l: linlC lIllllS 'i'~ = L:..I ~ T M ' 1¥i =- Lii~+Ml i 'I'~

=;

L~t:.

(2.'3)

+ /If f:"

~ = L~j , +MJ i

into (2.t2 ) ....e arriv~ a t tbll energy eeeveeetc n equa tions in the GOOrd hl-lt e syste m a, j!" el:pr85lled in te rms of currents . In ioiag from t he oon t ranslormcd coord ina te ays te m to t he s ystem a , j!" we s hould refer to Fig . 2.1 a nd dotermi ne t he pro jectioos (If roto r vo lta,ges and cu rrents on the stator u :is from t he relatiol1ll

u;.

= ~

ui ,..

coe 6

-II~

+ "' s in 9

si n 9 _

j~

~cos 9

t~ =

-~

si n

u' eee 0

+ ;' sI11 9

a + t; C,O!! a

(2. 14) (2.15)

The tr ansfor ma tion matrix he re Is

G ~ I . l)~S a Sinal ....,.gm G cos O

(2.1G)

T h e iOVOl1l8 t ransfor ma tio n matr tx Is

o-

l

eos o -sin 91 ... s inO c089

(2.1.7)

AJJ mentioned abo ve . th e ro(orellc8 fra mo with t he ref erence 1lXEl8 d and q rig idl y co nnec ted to th e ~otor struc t ure is very popul ar. 10 t his syste m. (I). co w ,. )~m (2. U,I) il foll ows tha t

lit "'" R'!' :+ rfift'ldt -

U'

-

-

+ jfiJ. if·

= R' J' + d'V' ldt

(2. 18)

Resolving t he ruoultant vectors alooa !.he d Iud q DCS yieldl the equl tion!l for the primi Uve ma chine in t he coord inate sys18m d . q; ~

_ &~+ dlf;,ldl -tal,'Y;

u; =

j~~1f- d'¥~Jdt

itd-

j~r:i +d1J'";,ldt

U; =

I;r; + dll' ; ldl

+

(0,

lY~

(2. 19)

2.1. The

E qu a ~ o ns

~9

01 the Gen e raliu d Electric Machine

Repl aci ng the fl ux linkages by currents . inductances, and mut ua l Inductances, we der ive th e direct- and quad rature-axis equ ations expressed i n teems of currents , in t'he sa me way a$ we did in th e a..~ coordinate s yst em: r~

u,, u'

•

1t~

I

+ (dl dt) Ld

(dldt ) ,If

(d fd l) M

rd + ( d.'dt) L~

o

0

MIJJ ,

L'dMi , + i r + j{oL1 i a

i ,r,

(2.31)

2 2

Here / 6>[, ' ;1 = jOl (M

+ t" , ) i , = j6>M i, -!- jool", ;,

j6)L2 ' , = }oo (M

+I" ,)

Replacing va riables

U1 =

f ,, '" i ) +

' 2

i,

=

j(jJ:11 I~ + f Oll ~2 ' 2

(2.3 2)

in (2.32) g i ves

1>,+ fwMi, + ,,;4- joo.11i , }

55

bra nch includes t he res ls ranc e rG equ'ivn leflt to th e i ron loss. AlLer t ransfo rmations , Eqs . (2.41) are wrj tten !IS

V.= R/. + ix/.-it. . . . 0 = E;-

(2.4.2)

jx;n- R;f ;l S

i o = i .+i; Here R ; fs = H; -+- n ; (1 - s)fs. Int rod uc iug the s t a te r and rotor Impe dances s , "'"

R.

+ /z. ,

z; = R;

+ Ix;

yields th e eq ua tions for a n lnd uc t.iou ma ch lue

V.= - 80 + ;.:. 0 = Eo- i;z;- i .n; {'I - - .

(2.43)

S)iS

/0 = /'+ / ;

The equi vale nt c ircu it nnd t he phnsor diagra m for t he a bove equa ti ons appear ill Fig. 2.5 and Rig. 2.G res pectively . A cirde

R,

-

,;

x

'.

I,

U.

J.

I '.

.; -e-r-

R'r(l-s)j s

I,

Fig. 2.5. The equivalent circuit of lin lndu ct tou mach ine

diag ram ex ts ts for a transform ed squlva lent circuit. T ile ee mp lax equat io ns (2.4:{). equivalent ci rcuit; and circle diag ram are t he basic el ements of t he t heor y of stead y-s tate operation of ind uc ti on mach ines. The Sleady-sla to eq uation s for ay uchronu us mncnruea without a damper windin g res ult, fro m Eqs. (h34) for t he pri miti ve mac hine. T he eq ua ti ons for a synchr o nous mecntne t hat do no t tak e Into accoun t the dum per windi ng are too a pprox imate (t hey are given below follo wing th e d iscussion of mul tiwi nd ing ma c hi nes]. It should be borne in mi nd tha t the ph aaor- d i agrams of synchro nous m achines are drawn for Iafrl y si mpli fied equations : those d i agr a ms ha ve a qu a l ft at.ive meani ng for mos t s ynchr onous ma chin es.

cs, 2. Eht ch omeche nicel Ene'GY Co n~ e,sio n

T he ph nsor diagr am for cor resp ond s t o t he equa tio n

II

nocs nli ent- pc!e machin e (Fig . 2 .7)

(2.44) wh ere I~ is t he ar mature current: U is vol t age; E is the upen -eh-eut t vol l llge or em f; T " is the armatu re res istan ce; x . = x" :I:Qd is th e

+

E

I,

Fig. 2.1. T he phewr diallram vf a llon." li~ll l- r",le s~'n dl r - ;t.,, ), I'h e hlock rli llgrn m for E qs. (2.49 ) nppon!'g in Ap peudi c J. T he se t up (){ Fig. A2 il' stoblo , t ho nei lll i" e reedb ack sig nnl exce eds th e pos it i ve feedba ck ~ ign ll l, 81ul !If g(>rlern l lOIl j ~ ll lJ~en l. T he uc t et.len of th e eq uat tous for rUrt'61U8 anll fl u ); Ii nkllges: ta c hos en suc h AS 10 be con venien t (01' t he aimuln tiou proce d ure:

d'l';, . base momen t M b """- P b1wb. fl ux li nkage 1f/> ... Uh 1lUb . tim e t b = t /(Jj b, imped ance = 1> = u />li l>. in d ucta nce L" = :f.,,/(il b. and mome nt of inerti a J b = M bIro/>. The m nt bcmeucet model of an energy conver ter. where the differ ent i al equ ations fot flux Hnkage e are wr it len in qu en u uee expres-

sed in dimenslonless units, has the Ierm d'ir;'!d't ""'~lU~ -

a21jf~ + ~3iV~ , d 'ir' idT = a1u', dlfitid't = -a,'Y!: + ~1f~ -;",1jr~

. , dlV ~ /d't = - ay\jf; + ~ ,g1f l + w .W:"

-as~rl + aBo/~

M . = ~ , ( 1jr~1jr ~ - q.~ q.~)

(2.5~

dw. idll= ~'1 (M.- M.)

where

,

al

a , = u, =

,

R'L'

'

,

R'M

= a. = 1; a~ = a, = LlL' M . I~; a s = a.'" L'L' ft/' fI ' L' L'L ' !It'

,

t ~ ; Us =

.

1tJ.-

R'M

L' L'

t il;

• .iP I ,,; a ll = (mP n/2) X

a;1 ...

IM /(L'U - M ~) I (II't /1I1 b); PAM ,,/Jw~. Th e bl ock diagr-am for (2.53) t o be solved on El com pu ter remai ns th e enme as befor e. The sol utio n of equatio ns in rela ti ve uUtts enab les handlin g t he pro bl ems which otherw ise (when using rea l units) lead to unsta ble models . T he model of an en,e;rg y conve r t er si mulated a ll Oil analog computer re pres ents a ec nt rel syst em cont ai nin g posit i ve and negative feedbac k loops. T he degr ee to wh ich aile feed back path or an ot her errecte th e st ability depen ds on t he relatio ns between t he para meters of t he mach ine bein gmod eled. A si mul ati on model Is chosen proceed ing from th e ty pe dl energy convener. i ts ch a racte ris t ics and t rans ient beh evfor. In Appe nd ix I are g iven t he equa tfc ns And t he bl ock dia gra ms of th e main t ypes of ener gy converters. X

1.4. Transient

Pro ~esses

In Electric: Mac:hlnes

Transi ent s in electric machi nes ar ise from ch anges in the voltages And Ir eq uen ctea at ma chine te rminals, the l oad 011 the shalt, meehtu e pa ram ete rs, d uring conn ection of II mach ine t o 01' its di scon nect ion

"

trom th e bus , e tc. III re nt cc udh tons , t rllos ien L peccesses COli natu ra l-

I)' occur during II s lmu h a llOOus venntjou of • Ie... fac tol'!!. T he combin .tions of t ile fact ors affec t ing the d )'n ltm ics t a n La mani fold . so t he rese ar d.lor must h ue ollou gll es perieuce lind kn ow led ge La ch oose tb e pre v. i1ing se t 01 JIICI Ors. l h.ere.'b y aim pJit),jng the problem. The re ia II Itre " t " Bri el y of l.rIUISicllts wh lch are mIlCh. more ('ompllU t h an s!.ead)'......t ete proc esses , Ih o IaUor bc ill!,; • particul ar c ese of t he for mer. B y t hei r im porlllllco find the tunuea ce t ho)" ha ve tlll l b, cpe-ett ou of mfl on t he tn vesuRat io", of dyn ami c processoa b y m ean s of com puters: many probl ems lu v(> become ctesstcat an d fe rm part of t he h borotory wor k us i&,oe d to stude n ts in mos t «llleges. Figu fTll 2.10 t hrough 2,13 ill u"tra le the start oecllJ ograms for motors u Di ing i ll PO"'''E!f fro m 4 W to 500 kW. As aeen fr om t he oScillognms. the st4rtiDi p rocedure pattem d iffers wit h t he relat io ns between the patllmeteTll, T he }' A,U-22 motor GOmes up to Its s taad y-s t ale velocity

Ch. 2, EI..cltom.. chaninl Energy Conw" "ion

for tw o or t hree peri ods, hut th e rot or yet oscillates a bout its steady angu lar vefccit y for some tilli e.The 50o-kW motor gai ns s peed very alo wly, but it does not overspee d «nee approachin g the s teady-state veloett y, T ho star tin g condi uoue for motors of th e 4A series, for exam ple. 75-kW moto rs, a re most typical (Fig. 2.12). T he process of rO VO l"!:IllJ differs f"orn the process of s tarting by t he e ffect of pe ra metera on th e -hu pact torqu e M ,m. impact current

I 100 V

1= }

f'i!l. 2. ' " The o""ill ol:"mm of start l o/l t hl! I" AA-I03,H typ e 2SO -W Ind uetion mowr

I

. h ~ ..·

~

,

100 V

.

Fig . 2. 12. The u$ mo gram of stlHl.lng the 4,1, -250504 t ype j5-kW induotioll mow r

,m.

1 and t he s tar t ing (speed up) t ime t. , (Fig. 2.14 ). For i ts reversal, a j'raotor is c ut off and then 'co n nected to t he line wi t h the reverse pha se sequence. T he t rllns iont hero depends on t he commutation t ime a nd o n wheth er or not t he fie ld In t ho a ir gap has died out. \ Vhere t he swi tching is ins tantan eous, t he processes nf field decoy and fie ld buildup go o n concurrentl y, so the im pact cu rr ents and torques grow. T he sche ma tic ct e program unit for reve rsi ng is s hown In Fig. A7 and Fig. AS. , In restarting a motor, t he 'i nrushes of current and the t hr us t pro-d uced are t he highes t. T he schema uc diag ra m for restar t and the

! - OIl 1 8

"

s ystem of eq u ati onjll for t his e e nd t uon a NI t he se mc as at !-l arting with th o dlHeren ce t h a l the in it i al vefoc t t.y lolr is ot h er l h lln zet O> (F igs . 2 . ( 5 , 2 .16) . T he IIH\I Y!lis of t eet lr t p receseee i n voh' ing ~ an

o

" Fig . 2.t4 .

1', _ :

~

•

s w . ... _

~

0

I'v

_

I

l'~

,..

-,.< 0;:::

•':1

nl!ye raill~'

Nn

t he '\Jl -3t1 t-.::

-l- --4

6

8

10

t."

12 J

Fig. 2.19. The curv es or Al l"" and IfI>< "cr~u & moment of ' " cr UIl

o

V '" •

0.25 0.5 0.75 J.O 1.2 5 1.;,

H g. 2.ID. The curves "', ' .1 ' M r"" and lim verS\ls roto r leakage indu ctance

and 2.24 display the plots of tho sa me q uantities 3 S runeuons of mutu al indu ctance M , momen t of iner tia J , lind rotor leakage induct ance I~, respecti vely. T he transients in reversin g lire more complex t han t hose a t s tar ting. It has to be no ted t hat t he effect of parameters on t he processes occurring ; in reversing differs from the

eUeet th ey have on t he processes ahtarling . Op t ltrW m po.rdmeursln Jtor tf ng , n versing or other dy ntlmic condUioll8 dtller f rom one artother.

, •

s

e

,

0

" •

I.

•

,•,

,

r:

, ,• ,

1,.

J

z

, ,

,

1

"

o

Fic· 1.2 1. The eulYU of "I' I,,,, ve nll. rotor ~tln«, ..

,

,

M,,,,, aDd re~na

,.

I'--

,

'" I

0

2.

6

1

10M

Fir. 2. 22. I'1M wn'" of 'ot,

M .... and "'" ver$\l$ lDut ua l .IDduet.a1lCC in ",,-er· al Pg

,

e

•",

I

• , 8-

•• 0- 0

,.

0

1/ J

a

," • ~ , 8

,

I

0

,

""

,

'"

I

o

,.

M 8

0

'"

,,,

j I.

2

,

•

'.1>

,

00.25 0.50.15 1.0 1.25 t;,

s J

Fig. 2.23. The eu!'Vfl or M .... I.lId I,,. ...~l'SUi ml)mellt of ia ertUI te reYer·

,

Fir. 2.24. T he eureea of '",

f,,. vc ......

&I,,. aa d

.",..klp induel.l.QOl!in revee -

"·8 "'" is of interest to l ook al t he pro cess of reversing with t he field

h n ill und a mped. With a ra t her fas t reversal af ph ases a t ma chine term inals , t he li eld in th e ab ga p hB.!l no ti me lo give up its s tored energy. In other words, ' he fiel d bes nol yU decay ed a t tile IDUant

3.1. Infinite .... bil'e'y Spectrum 01 Field . in Ai' G a p

71

of cenoecuon of l ho moto r to t he bus. The anal ysis of the proces s of reversal t h us requires estimf\ting the li1i ti al values of the undam ped field by defi ning t he flux linkago tjetween t he st ator and rotor windings. Revorsin g coutecto rs o perate in tho severe conditions (an ec con rector c loses i n 0.03 to 0.05 s]. The most ellecrlve techniqu e of studying transients in t hese dev ices' is to use an ana log computer set up for prog rammable work. . One of tilt! interesti ng tr ans ient modes is the mode a t resta rting. The s ystem of equations and tho simulation mode l fo r its sol ution will be the s a me as at s tar ti ng with t:J1e excep tio n t hat the angul ar velocit y 00, will he oth er th an eero. ~ ll v es tig ati on s reve al t ha t t he restar tin g process at w; close to t he nomtnal (with shor t-time su pply interruptions ) is accompa nied by heav y surges of currents a nd torques. Th ese s urges exceed t he ma ximu m val ues of euerents an d torques in etartfng and reversi ng. The restar ting process wi th th e field undam ped len ds itself to t he anal ysis af ter estimat io n of th e initial cond itio ns for [lux li nkages. Since the rotor has a s tore of kine ,tic energy and th e Held has not decayed. rest ar ti ng mus t appear lit fi rst glance to be an easy mode for an electric macWne. However s ince restarting Involv es th e d isconnection of II machine h om and its connection to t he s u pply line, tile two attendant transient processes witb complic a ted changes in currents and to rques s uperpose one on th e ot her . Giv en the mat hematic-al model ,of a n in duction mach i ne. we can ana l yze tho opera tion of the machine bot h in t he br aking- and in t he gellOratlng modo. In inv ostig-~ ting the gene rator- act ion. i t suffices to change only the sign of t he to rqu e in ui e elect romecha nical equat ions. If t he line voltage is cons ta nt . t he ind ucti on generator operates in parall el wit h and into the infinite power li ne. Eert.aln difficulties anise in the ena lyaie of an' auto nomous Indu c t tc» generator when it dr aws the ronctive power ~ro m ca paci tors and t he voltage and froque ncy unde rgo c hanges.

Ch apt e r

3

General ized m-n W inding C onver ter 3.t. The Infinite Arb itrary Spectrum of Fields in the Air Gap As is 'known , t he cir cul ar field in t he ai r gap ca n be tho ught to exist only In an ideali ze d machine. I n real ma~htnt8, the air gap u hi bUs all in fini te spectrum of harmonics dif/trln g in amplitudt ana

Ch. 3. Gcr,.. •• /i zed Ill ·a WJndlllj Co nverll!l.

fr«JlUnclI along with the fundamental harmollU:. T hese har mo nics re volve both i n t he for wl ro and in the ba ckward directi o n wil b res pect to th e revol v ing fund amental harmonic. The angul ar veloci ti es of the harmonics can be h ig her a nd 10"".r than tba t of the fu nda· mental wav e and their amplitudes can vary in ro ta tio n. All heemoflies IU Y be d iv ided i nto two types. t i me an d s pace barmon ica. T ime Juumerna are th e ones whic h let i nt o ~he air gap of I machine fr om tho ou ts id o. Spa~ harmontc$ a ppear in th e ai.r g ap on ae-couot of the specifics of th e con verter's in t ern ol struct u re. I t sho uld be kop t In mind Iha t thi s cl assi fi ca ti on of b erm onrcs is rathe r ec ndlt ion a1; the Dames ' t ime and s pace harmon ica ' ar ose for h istori cal reasons in the course of development of t he weory of el ect r ic rnaeh i n ~ ,

Consid er ing tho Ee 8! /I t \l'o port (see }' ig. 1.8), we:sho uld not e t bot the conver t er has t wo in put s , one on t he s ide of e1et:tr ical teemino!.!! and thn othor on tlie s ide of mechant cei term in nis . T i me har monica arise from nons tnusoldel, as ymme tric Yoltag!!5 and nonlinear cha nges i n the a m plitu de and f req ue ncy of v oltag es. T ho)' also resul t f rom nonlinear changes in th e to r q ue and s peed. In the general ease , t im e ha rm on ics appea r from t he simu ltaneous aellon of nonlinear f ac tors at t wo in pu t termin als. T hese harmo nics m ay etec ieL in to th e ai r gap of an elec tric machine fro m t heMnal ter mi na ls (sec Fi g. r.t }, Heat s hocks (sharp t em pe rature nriati(lns of t he machi ne. frilme) cause u pper barm(lnics in th e a ir g llp as a reo6u lt of cha nges in tb o machi ne psramete rs . NOll5in uso idaJ YoU,ages whtch giv e r ise to ti me harmo nics ma y res ult from nonlinear e lements s uc h as sat l/ra ble reactors and semico nd uc tor el em ents d ispose d ah ea d of the mot or, a nonsmuaotdal waveform of the generat or vol t ag e or d ts tce uo n of t he wa vef or m of t he su pply vol t ag e, etc. If th e su p pl y vol tage eoetatn s a ccn etent co m pone nt , a harm on ic spec tr um emer ges , w h ich Includes a n in finite r/lnge of even har mon ics along wi t h od d harmonics. I n t he a bsen ce of urn e hai mo n ics in t he air g a p of 11 machin e, s pace harmoni cs originate fr om t he nonsi nusoid al dt et r tbu uc n of tu r n!'! an d magnethi ng rorces'. air g ap nnn -unHormi t y due t o t1le presence of teet h and sl ots in t he ro t or an d s ta tor, gap eIJipticity and C(lnic ity, and non lin ea rit y of t ho peeem etees en le ri 0il' te te el ectromech anica l eq uat ions. Consider in mor e detail s pace harmenice, in p nrtieulllr t ill.' harmonics of mag netizi ng Iorcea, The wiDdi ngs of elect ric machines a re cu rre nt l oo ps producing magnet iZing rorece . The s im ple:st windi ng (loo p) is a t urn (or II co il consisti ng of several t Unis) wh066 pi t ch II is equal to th o pole pilch 't (Fig. a. l a) . Such a l oop p rod uces a rectangu la r mag ne tom oti ve force (m mr) . A t 11 < 1: (Fig. 3.tb) t he m mf takes t he form (If II trllpezo ld. W here t wo or more ccus are Invo lv ed, Lbe mmf llSS u mes t he form of II. s lcpl iko cu rve (Fig. 3. Ie ).

73'

3.1. 11l/lIlUe Arbil rary Spect, ul'l ' 01 f ie ld . in Air Gap

Developi ng the mmf as a harmonic pr ogress ion , we s hould note th efact that where t he mm f distr ib u ti on is rectangu la r in sh a pe , u p per harmon ics have maximum em ptl t udee ] th o am pli tudes become low erwith II sho rt ened wind ing pHdl /lnd lower s ti ll fur th er where t ho

,.

' oj

Pik. 3· 1. Curn-{\t looJlll

winding consists of II few coils. Only ill the case of the si nusoidal' distributi on of turns over a smooth cy li nd l"iCl,I1 su rface of th e ga p' upper har mon ics of the rmnf are no nexletent. F or a co il Windin g or Ior one tUPi, as s ho wn in Fig. 3.1a, tbeampli tudes of harmonics are g ivell by }\ = (4/n ) F c F~ -= (113) (4{n ) F e F~ = (1/5) t4/1t) F . (3.1 } F \" = (1Iv) (4/n) F e where F c = l wl2; I is t he cu rren t Clo'wing in a t urn : and w is the number of turns in t ho coil. In a willding comp r ising lJ coils, th e m mf epprextmetee a si nus oid and the amplitudes of har mon ics become lower. In t hree-phasa symmetr ic wi nd ings t hnre ap pear ha rm oni cs of the or der

'\'= 6c±1

+

where c = 0 , 1, 2, . . . Ha rm onics of th e order 6c 1 (1, 13. H) .. .) revolve in t he forw ard d irection with res pect to t he rotat ing iifllt har monic a t II s peed Wllich is a factor of 7, 13, ii}, . . . l ower than the sp eed of lite fi rs t ha rmon ic, ;Harm on ics of the order 6c _1 (5, 11 , 17, . . . ) revol ve nt. n speed of 1/5, 1111, 1/17 . t hespeed of the firs t ha rm onic in t he u troe uon o pposite t o that of t befirst harmon ic.

74

Cil. 3. Ge,,,II..d 1lI-n W ind ing C a nvetler

For two -p hase s ymmetric wi nd ings , v = 4.:- ± 1 Harm onies of ·t he order 4c + 1 revolve in til ll same d irec ti on as the first ha rmonic, .an d harmon ics of t he order 4c _ 1 revo lve in t he op posite di recti on "t o t he firs t ha rmonic. Because t he phases of wind ings are aaymmet rtc, each upper hnr"monic of t he mmf ma y have a for ward and a backward wave . T hus , -oni y .'i th t he gea er al t eed elect ro mec b an ical en orQ'Y ec n\'cr t er- t he m ath ematical m odel of 8 real m achine-wh ich IDll}' hav e ma ny loopl'l (phase win d ings ) on t ho sta t or alld rot or.

77

Such a pe neralieed co nve rto r eDables, us to wri le eq u aU Oll!l wi t h d ue rer ard lo r t he iield )~D t he a ir g ap and aU cUlT9nkarry in g l oops. Sum ming lop. th~ ~MralUed ~nugg con t~r'~r u an id=U:ed twopole tuXJ'pho..u: ( ketrle machine wUh Ili-n wind ings 0 /1 the slalor and rotor, ,.,spect llMl y, 4rrtlJ1, ed along 1M a "ami '" 41 fho lDn in Fig. $ .2.

au,

u'.....

I"

""'i) ..... ..J) OIl•

..

i .. """i) -t' W ••

u'

u:.

w;.

Usi ng t be mod el of tho gene rali zed converte r we can desc r ibe s ym me t ric m ul t ipbllSfI m ul tipole r ffiaj;;h in os on t he ass um pt ion t ha t t hey e re t rausform ab le t o a n eq uival en t for m l.o matc h t he t wo-phase two-p ole ma chine. As seen fro m Fig. 3.2, each phase wind ing: nee ne d eslg n aelom eubecelp ts Cl: lind ~ iden tif y t he ax es: a long wh ich wi ndi ngs w Ho; 1, 2 , .. . , m. n s tlUld for t he ordinal n umb or of t he wind in", 011 t he s t a tor e nd r ot or reepec rlv el y: And s u persc rip ts a a nd v den ot e ti le s ta t or an d ro to r win d i ngs . 1'eSpOctively, eu ppl led with vo l tages u . E ach p air of t he wind in gs is fe d fro m a n ind iv id u al s up pl y sou rce or a ll wi nd ings a lTllnl:ed in an y kiiub of ne t we rks draw cu rren t fro m a s ingle source. In the gener alized co nv ert er. mac net ic li nk betwoen t he groups of windings on ,the n me lids m a y DOt e:ziat.. E ach pa ir of windings (co ils) on the s ta to r prod uces a circ ul ar field i n t he a ir ga p. As is k n own, tb e generaliud convert er is an unsa t urat ed ma ch in e , whi ch allows m to use t he princip le of eu pee pcaitton. The field in the a.ir ga p ca n 'be .set u p by a ppl ying to t he win di oirS t ho voltag es of d iff ere nt a mp li t u deS and freq uencies , sh ifted i n

78

en. 1.

G...... . llted ....... W ind ing Converter

ph ase wr th respec t to eaen other. From the vie wpoin t of m etheDlfltieal t heory, eloctri c m achines d iffer from one anotbo r b)" t h.e {or m. of t he fiel d in th e a ir lap, n um ber of windings, an d the pan met ers of wi n dings . . Tho gt!ner ali f.ed co nver t er is a useful tool for descri bing allY elec t ric mecnrne. Fo r ex ample, I single-phase s ioQ:le. wind iog motor wi th • pUISllling field in th e aIr gnp ca n be r ep rQ!len led by 8 malhema ll ul m odel com pr is ing t wo pa irs of windings on th e sta te r and t wo-

Fig. 3.3. The lM del of a swgle·ph u o .aoto.

w:",

Ie:

pa irs on t he ro t or (F ill'. 3.3) . :W i nd ings and ~ build up n forwa rd (p os!Uve--sequeoee) fia l l'l l (lt~.. = U'" si n (,)1, ~ = U.,. c os wt) . Wi ndi ngs W;Ol a nd w:~ are t fed wilh voltages u~ .. = U", cos wt and u:JI - U... sin OJf which 8 8 t up a backwa rd (negat ive-sequence) f ield . It the rotor is of t he kquirre ), c8go t ype , u~ u~ D' an d u; ! are eq u al to zero . By chang ing t he vo hag(\(lJ across th e wiod lnp which prod uce Lhel orwu d a nd ba ckwe ed fields, i t is pceatble to e o oV H fro m t he pu15ating field eo t he elliptic. fiel d an d t hen t o t he circular fie ld if lIle blckward fi eld in t h e ai r gap does no t ex ist. II, a part from the forward and t he ba ck ward fiel d , upper h armon ics are present in the ga p, It is necess ary to a dd I req uisite nu m be r of p ai rs o f wi nd ings to t he mod el and l!illl u p t he fi:eJd b y ap plyi ng to the !il at or wi n dings t h e vo ltages of co rres ponding a m plitudes an d Ireq uences , whi ch sh ow a defi n ite ph ase sequence an d p h age s hift.

It:

..,

u;...

7.

3.3. The Equ. llo ,," of the Ge ne ..lb ed EC

i\1 0fI ~ cteewre mac-hi nes ha ve sev eral wind ings. If eddy current loops al'6 ta ken into eecount, :III m ~ i n es may be t bought of IllJo multlwind ing mecb lnes. As menlioned earlier, the genera Ut.edconverter serves as II powerful tool for t he llIIol )"ais of act ual mach ines . wH h many win dings. Equ . t ion5 describi ng t he bohavior of th e m.jorl ty of elect ric mac.bl ncs can be s et up using t he e;quations for t he genenlbed' encl1'Y converter. For t hi! we need to. expand t he mml of th e field (whose shape i n t he gllp Ls known) in t.9 a harm onic seri es, cons t ruct t he model from 1I few pairs of wind illils and ap pl y th e vol tages or corresponding amplit udes lind fre qu enc ~es to th ese windings . Alt houg:h tho fielll pattern ill a rot a Hng m ae~ n o is pn ctl cally impossibleto defi ne, the mauiemauce t mod el efl e u oncrgy converter enebleeus to 50[V6 mllny probl ema to a sufficie nl accuracy by s.pecifyiog \' Olt flg C1l on t he input termi nals of the conver tor. Th, onaly$is of working processes In electric machines relit:! on two" sttll-tm, nts: (a) all Slfl U C and dynamic Cha racter/d iu o/a machIne are' governed bylh.e proc, sses occurring in the air gap : Ib) an eltaru: modl l n~ is represellttd as Q systtm oj li near dectr ie etrrulu 1l101'ing with rnp«t 10 on, onothf'r .

3.3. The Equations 01 the Generalized Energy Conve rter The v Oltllg1! eqn a uo ns for th e gei:l~r a.l iud eneq: y een veeter ant written ill th e form of a eomple:a: ma trj x sim ilar to t he K roo mlll rb: for t he prim il ive machin e of Fig. 1J 11:

,

u~

""-

"'

u,

-

A:" A:." 0

0

D.

R••

B• • D. 0 0 A&'

AI

A ~' A ~

A,

A,'

x

III,

/,

(3.3 )

Each elemen l in tb e eb e ve mat rix is .. subm U rb. Rere and ere mnt rtx columns :

-I.

u:" = -I.

-:,.

u~~

1.1;. =

-I. -:,.

U~1l

u. =

-I,

. ;,

u:.. u~ . ",.

u~ 1l

u; =

u ;1l

":..

(3.' ),

eo

Ch. ), Gonerllljzed lIl_n Winding Co nverter

Also, I:",

I~,

',. il

.

are matrix colu mns;

i~ =

,~

Ii;. =

~

il,e

'" Ii .... ih

ih i& = .,

,

I~

':"

[0

,.

.,

~ ;a

'l"

'"'

Eqs. (3.4), It:... u:.., . . ." u:..c.: u~ .., u;".

. _, ur. ~ ;

u: D. u: e, . . "

u:.,. ~

.,

(3.5)

'.,

u~,,; u~ e- u~-"

...

are t he s t at or and rotor volta ges alc ug thea and ti axes. In Eqs. (3.5), t:", a; I ~ ... . . _, i~ .. ; lill ' t; D' . . ., l~~ ; 1;/1, 1h . . _, i:"s are Ute currents .a long~lthe a. and ~ axes in the stator and rotor. Volta ge equa tions (3.3) may be written i n a mor e gBllCfal form

i:

Iii] =

1z1

X

i;.., .. " "..

t;... . .

II)

(3.6)

The Impedance ma trix [zj incl udes 12 sub me t rtces. Four impequllt io ns from i3 .3). In these aq uaU :8 are the ste tor voh a ges At t he Iundam unta l Ireq ue ncy , \iODS, lind are th e stator \"ol tages prod uci ng t he fie ld of a thi rd har mo nic . Th e laller vo lta ges ha ve a fre q ucnc~' f . = 3/1 li nd li n ampli t ude t hllt corresponds to the IImp,li tude of t he fiel d of the t hird Rarm on ic. \ 'ol ta ges u;... U;, corrt"spo nd to tile fifth harmonic of frequeocy /. _ 5/1' a nd voltage' u:..... u:' 1\ are t M vcl tegee of th o m lh ha rmon ic., f .. =- m/I' The phazea a nd t ile dt eeeucns of ro tat io n of t he field har mon ics res ul t from th e cortE-spo ndlng vol ta ges pcross the sta tor wind i nas . If the a ir gap u hib ils s ubharmonics. t.e . theCields whose frequ enci es are betew the Iund amente l rreq uency . II pllrt of th e wi.ndi nga a re fed wit b voltages at freq uencie s lower t ha n.

U: .

u.:..,llu:.!

Ch. S. e .... gy Co nv"rslon .1 l'olonsln. .. AIyl'l: Vo lI!glll

'!Il)

1I11:~ Iunda me ntel . If the vol tage con tains even harmonics . a par t of t he windings a re euppljed wit h vol tages displaying eve n harm on ies. For t ach p... ir or 'rindi ng; on the s tato r t here is a corresponding pa ir of wi nd ing! on the rot or. IU rega rds an ind uctio n machine . lho voltages ac rOM t he rotor win d intrS are eq ua l to te ro for a shorL- t

vs

d'l'~

d '¥~)

-T.~ +-,- " -Z"BC + UA8 t L d l • e + L. It.. = 3 = r • •e 11. --;n- --r:;X (

(

X

(5,3'}

+

_..!..

d 'l'~ _ Zdt

1/ 3 d'l'8)

(5,32)'

2 · dt

Eq . (5.32 ) c a ll be dt soarded since for the phose c we beve It.. = - (It ,. + al b), f H = + l.b)

-«...

For rot or ci rcuits th e equ ations assume the form 0 .... - r. (L rnIL , ) I." (r,I L,) iV& + do/&'dt + ltI,'P6

+

(5 . .'l3}-

0= - r ; (LmIL , ) [(2t.(.+ [",) (Val + (" , IL, ) '1'$ +d 'Y ~ldt _ IIJ ,'¥;

(5 .34»

The elect ro mag net ic torque eququc n is given by M

- ..!. 2 P

1-

Lm Lr

[1f' (:<

(21.;,, + t. ~ l

'V 3

- 1f"ill."]

(5 .35),

A s imilar a p proac h ts a ppr opri a te in deriv ing t he eq ua tions for t hoind ucti on motor wil h t h~'ris tora ewtt c ned t o ot her mod es of operat ion . Figure 5.7 s hows th e sc hem atic di.a.gra m Hlustre tlv e of t he wa y of connect.ion or the ere tor wind i ngs 19 't he s up pl y l lne via se ri es-co nnected t h yristor cells . when on e of t he th yristors i n a t h yr istor coli TC ie hol d in th o cond uc ti ng s t ate (~'he s wit ch is 0 11), the thy ris to r ('.ell -osrstan ee is close to aero ; when' it goes t o t he noncond uc t ing Slate (t he switch is off), tho t hy rist(!r'cell has a resistance ex tending t o infini t y. B y Kirchhoff's second law ttllC

+ leR e + I"R A -

uo"e. U"e~ -

t"R n = 0 t.R C = 0

(5.36 ), From t he se t or eq uations (5. 36) We ca n deter mine curren ts and! t hen vol tages in t hy risto r cells: _ I R _ " ", " c (IiAIl - U. 1th) - ( UC"' - uuc) RA"n (5,37} U CA

UT A -

c

• -

R/iRc+ RARs + " c"s

Ch . S, Ene rg y Converllo n at No... ln. .. AIY"" V oltage,

1.t 2

S i mil arl y, we ca n define U ra a nd UTC' T he eq uation! for u r A , a nd U T C perm i ~ lIS to determine th e voltnges across t hYt'istor ce lla ope ra ti ng in au y of t he s ta tes of con d uctio n. These vo ltages are n~ ar y for formul a t in~ t he programs to be r un on di gital co m-

U r a

pulers. 10

devolo ping a ma t hem atical mod el l or t he s i mu la tion on

8

d igi-

tal co mput er, i t is desi rable th at t he model sho uld most closely approxi ma te in s t ruct ure t he roal

erreuu (If

Ole th yristor d ri ve und er

- , "n

i~

TCA

B

~

" ><

~ TC,

~ TCe

u ,~ 0, Y A I = 1 (Y" t = 0 ); U i~ < O. Y " t ,.. t (Y" I "'" 0). 11 t he t hyristors enter th e ee rc-eeod ucucn l'\lllion and th e funcnon A OJ Is eq ua l to zerc , so t ha t none of t he t hyristor triple ls s wltohes on, we need to kno whe t her t ho t h)' ristor'S t ur n on pairwise:

+

+

+

+

+

An -

+

+

+

X"lX Bt Z UZBl + X AIX C .Z A1 Z C ~ + X A. X 1HZA 2Z BI + X A1X C:llZA IZ CI + X 8IX C' Z B I Z Ct + X nIX CIZ B,Z CI

(5.40)

When A u _ 0 , t he system ma y golfrom th e aero to the two- phase conduct ion mode. Assume, for example, trigger pulses ar ri ve at AI and B250 l bllt u.,." > o and U'I' B > O, but UT" B = U 'I' A - U T D > > O. Th e system t heu s witches from the oU-s t Ate to t he state of tw oph ase co nd uctio n. Consequently, 1lp.llrt l rom t he function A ot defined by Eq. (5.4.0) , we mu s t consi de r lin addnlo na l logic function

B .. = X A BZA IZ S2 +

X " CZ~ IZCt

+ X . " Z • • ZA.

+ X CAZ C1Z At + X s cZ elZ Ct + X caZ c IZ s.

f5.41)

where X " B - I if X S " 1 eX " s "'" O}. At l.I'1'AS < 0, X BA DO "'" 1 All - 0) , and so on. T hill logic l un cUon describing t he condition I t whic h t he t h yristors are dri ven on for t he t.wo-p hase co nd uct io n bas t he form

ex

An

='

!!"I!! •• + VAIVC. + !/AtVBI + "A tVCI + Ytll !JC. + " e2VCI

(5.42) T hus, from t be a bov e discussi on t he followi ng co ncl usion c an be drawn. a -O l l tl

tH

Ch. So

e....'srr

CO...... .$lon f1 Monl on. •

Mym..

Vo ltag e .

fo r the three-phase cee due u cn mode rbere corres ponds I l ogic functi on C~ - A u An + An (5.43) T he logi c fun cfi on for th e "tate of two-phMe conduc tio n is C ~ _ (A .. AnB.. An) C, (5.4 41 )

+

+

+

- stands for th e negative ' fu nct ions of C ~; at weeee C, and a t C,,., "1 we h i ve C, _ O.

C~

-

... 0, C,

= 1

T he fu nction for t he s t l t e of zero conducti on is

Q. =

e,c.

(5.45)

Tho a bove l og ic al an alysis enables 119 t o pass from one set of dllfereo t ia! equa t io ns t o enctber depend ing VD the s ta te of condu ct ion of th yris tors lInc di odes. T ile algo r it hm for det ermln,i ng the state of t hy ris tors is so cacsen as to carry ou t, t he eeq ucn rtal ane lysia for t he t i me in terv als ov er whic h t he pulses on all t hyri.51ors remai n In va ria ble. The sYlt em can cha nge s tate ov er t he given lengt h of t ime if , th yr is tor tuns on (as U1' on t he gate changes sig n) or a t hy ris to r cell turns off (as: cu rren t ch anges s ign). T ho swi t ch i ng mom onta are so ugh t and the corres pond ing values of v ari ab l ~ n e found by di Vid ing tb e iotegrfltlon step in b u f down to H. 1n " . 10-1 s, th en t he Iinell r i nt erpo l atio n b perfo rmed. & th e pulses d ie out o r ne w OOCli emerge , aaothtN' criter ion is I n u p to de termino the presence of pulses on th yris t.ol1l ; the above-desc r i· bed method is t hoo used t o defi ne t he stale of t hyr istors an d t o conduct t he logiul a OI I)'sls of t'bei r opera t ion. Based on t h is calcul atio n technlque. a generali zed r ou t ine is writte n t o &nalYl e t he dyna mic an d s ta t ic perfo rma nco of t h'e g iven sy stem on a d igital computer for any character of ch Anges in the c.ondu ct ion ongl l"S of th y ristors,

5.4. Pulse

EI~dromec:hanlcal

Energy

~o n verters

E lec tromecha nical de and ac system, belong to t he continuous type. There are d iscrete electromechanical S)'s t oms where t he eonversion of energy proceeds by way of the pul ses of elec tromag neti c powe r. T he las t few decad Ol' h av e s een a cons iderab le ad van ce men t i n th e area of d iscrete elec t ro mecha nical syslf:D\I. Pulse ene rgy con verters ope rat e both in t he moto ri ng mode (ate p moto rs ) and in the ae neutin g mode (puls e ~enerators). Ste p mot ors tr enerorm vol ta ge pulses into discret e an gula r motions or s tep wise linea r d isplaeem en t and pUlse genera tors prod uce powerful curren t pu lse! fed t o el eetropb ys ical setups. Step moto rs have a s mal l power , us uall y up t o a few hu ndr eds of watL!!, and pulse gllDerat-ora are

SA. Pu lse Eleci,omed,.." ica l e ......gr Conv,,"""

ll S

gen er aJly b u .i l t t o g iv e a h ig h o ut put. u p to t ons of m ega wat ts in a. pulse. A s tep m ot or is m a de com plete wi t h a co rn mu t ating e r renueme ne and designed to c a rry a load on t h o r otor eh a tt. F or eac h mo tor t here' is a d efi nite s wi tch i ng freq uency a t w h ich its r otor- fol low s in step a v ar y in g fi eld in t h e ai l: gap . This Ir equoncy is blow n as a resp onsef requency d o termiued b y t he enure s ystem com p ris ing t ho sermcn uducto r co m m utator (pu ls e gone ralo r) an d t he step m o t or . an d a lso by t ile lo a d all t he s haft. Most step mo tors are multipolar mul ti p hase synchro nous m ech moa in which e ithe r g r ou ps of s ta t o r wi ndings or ea ch win d in g recelvoa unipol a r O f t wo- polarity puls es . T ho r otor re volves ncu u n ffor m fy wit h i n the en tire I\ngl e of ravof u rron as it follows th o step wis e d is plu ce rnunt of t h e m agne Uc fi eld. T h o m o to r lncc rp orates s pecial me a ns o r provision is m a de i n t h e m o t o r d03ig n t o fix t ho r otor in a def i nite position to rogisle t' t h e response . Cho nging t h e or der III ar r-iv ul of pul ses , it becomes po ssi b le to al t er the s ense of rota t ion of t hc ro t or Mid t llu:;l s u m \ I P the positi.... e a nd neg ati ve p ulses in t h eform of angu far dcnecuoos. Step electro mcchani ca! syst em s , l ik e all ol ec,l!'ir, m a chtn es , are con vertib l e. T ho }' can ac t us s ou rces o f low-power p UISM. A sl i lll por laut tusk is t o produce h igh-po wer pu lses u p t o 100 kJ wl th n steepload in g edge lind a lligh repeti tio n rete, or PU ISflS o f a defini te waveform . P u lse g en e ra ta l's must st ore e ne rg y i n ure form of th e k in e tic (' 11O.-r gy of ro tati ng me m bers a nd i n the form of t he energ-y of rh e m agn e ti c fi el d . O n t ile on e hond , 8 pulse ge nerator mus t. b e c a p a b le of co n t ro ll i ng t h e energy eto eed a n d , on t h o other. mus t h av e 8 sm al l time cons t an t to pr od uce d esired p ulses . These are con fl ie,u llg requ t rc me uts fr om t he ....ie wpof nt. o f elec teomochanlcs. Diffic u lties o ft e n ar-ise in d es ign ing a magnetic-Held ty pe pu lse ge ne ra t or a d equ ate to t ho r equirements i m posed o n it. It t.her-efo -e makes sense to reso r t t o p u lse g enera t ors of t ho electric-Hel d and electromagn otte-fiel d ty pes. Supercnnduett ng m ag net tc s yste ms orretmu e h p romis e as de vices for sto ring l a rg e e n cigtee in th e m agnetfc Held , Tho epproeches an d eq uat ions ap plied in t h e ano lyais of o l'd in a r y energy co nv ert ers also ho ld for t he s t u d ies of al ectrcm ech anle a! ener- . gy c on v ers io n processes ill pul se en ergy con ve r t ors . T h(\ form of e q un t tons fOf pulse vo ltll.gC.'! remains uie s a me us for s i n us o id a l v oll nges . I n t he si m u lati on of s tep moto rs it is common t o s im pJify equ atto na so as to t ak e i n to accoun t th e ch nr -ec t cr- o f l oa d a nd t ho in tcrll lli re efsr a nce of the s u p p ly sou r ce . Continuous s in usoidal s u p pl y v oltages ca n be t h ough t, o f Il ~ anin fini tel y l ong trai n of puls e.'! v or y i ng in ampl j tude. As the \' ollllgO'" deviates fr om th o s i n us oi d , t he ro t or rovo tves at n n onu n ifo r m ang us-

'"

l e e s peed; t he motor ha vi ng I'l defin ite form of the fiel d d ispl a''''"\'''. .. In d ln . : wofof lin d "11'1-

of· a nd

'I:'"~ IO

d Gm pe , w ln d ln;O l

"'I _ h lll d

"lllG H.1I"

dIng, field windin g, and dtJ.m per windi ng (IS shewn in Fig . 6.1. In t he si m plt!lSt case. 815)' l\c hrono~ mechln e is a tb re0-wind ing machi ne. its mod el h eing represented hy si x ph ase windings .

118

Ch. 6. M"tti .... lnClng ~o:hl n .. 1

Il is customary to ex press the equations for II e ync h eonous ma -ebtne in the sy stem of d-q cecrdinetea rig idl y conn ected to t he roto r s t ruct ure . For a ma chi ne with tra nsform ed windings we wri te down the foll ow ing cquaUoll!l

u " _ doF,i dt - 'I'"w,

+

+ r..l"

d'Vt/dt '1',,(1,). + 14./ - d"o/ldt + 71l f 0 ... dWd,idt r,U ' '' d 0 _ dll' ",/dt 701 tl ,, ,

14., '"

r.',

+ + H er e r", an d r, are th e fesis tnn,ces or the .o.rmaturo and fie ld

(6.1)

win d ings eeapect ivel y; rdd and r", are t he rostsr en cce of the d amper wind ing, r.e. t wo ph ase wind ings al ong t he d irect a nd t he q ua dr atu re a xis res pective ly ; 14." an d u q ere t he :voltagos ac ross th o a r ma t ure wind ing , t.e. t wo phase win di ngs alool t ha d a nd q exes r&pect ive ly; end Is Ula voltage i n t he fiel d win ding. T he m8.i'notic nU l: li nkages Ior the w i nd i ngs are 'given by ' .

u,

•

If'..

c:o

•

L oI ~ oI

+ M odi, + At . di ll"

+ M.q'"q 'V, ... L Jt, + « , », + iII. II' . " 'Y"" .. L I .'•• + .+1. - U bl (i) b. i m ped ance : 11 - u ,,!i ,,. and ind uc t ance L b =

....

+_ =1l".: +A

= ; b! CJ) b ..... :1:.. ".

U 61i 6 flJlr

Here i}f G,, -

ill." - z .. q. 1.." - Z", L , - %}> L .... - % .. " ,

z"

'" f"Tq

L, = and L ,,'l. In mOd eli n~ pc rm an entrmlgnet 9'1s. 5.2. The model of « 'f1Id u'OftOla _ hiDe ' ynclironou.s mach ines , t he dee-rea of exci t a t ion of a magnet is delined as t he p rodu ct of cu rren t J ... tl me.a t he mu t ua l Lo..du ctanee of t he a rma t u re windi ng and the fi ct itious curre nt of t he magn et. Pe rmanent mag nets a re re presented by an eq uiva len t inetti aless and Ioaalees loo p ellppli ed fro m a dc sou rce (see A ppend ix I , Fig. A6). W ha t co m pli cat es th e s i mu!at ion.ol sy nc h ronous machi nes la a eet urati on -ind uced chaege in the parame ters wit h load . In s el t en tpol e machines, th e d iapla cemem of th e fiel d nxls with respect to t he nu of poles a dds st ill more to t ha complex it y of t he model. T he t ran s ien ts were giv en tre at ment fi rst i n s ync hron ous mec hi nes, As fat bac k as t he l a le t920s P ark an d G ore v fonn ula t&d Eqs . (6.t ) wh ich now bear t.he names of P ark an d Oeeev. T he need for the s t ud y of t t ans ien ts in s ync hro no us machi nee a rose in th e cou rse of de velopmen t of power sup pl y s yst ems; t he aim was to in vesti g ate the effect of t he emerg en cy opera t ion of one O1ncbino on t he s tab ili ty of parall fll o per at ion 01 ot her llI,ach lnes. Since com pu ters c ame into being m uch la t er, it was Im posatble t o s elva E qs. (6.1) l or the oese of a vary ing ang ular veloc it y. Tho onl y wa y out was t o si m pli fy t he in it ial equat.ions to descri be the main e ven ts dete rmini ng t he beha v iou r of a mac h ina !in condi Uons co nside red most im port an t l rom t lla pr acti cal v ie w poin t. I n perfor ming th e anal )'Sis, it W all D&ce5!!ary t o define I set of param, rers cbarl cterb ing th e mac.hin e

%""

.-

".

Ch. 6. Multiwlnd lng MachlrlO'

o pera ti on. Th e focus of att$ t.ion W/lS largely on t ransi ents in th e condi tio ns of suddenly ll p~lie d s hort ci rcuits. With th ree-ph ase or asymmcteic short circui ts: pl aced on sta tor windings , t he s urges of curre nts i n t he windings excee d 10 to i s t imes the nominal values. 10 solvi ng Eqs. (6.1) wit hout the use of a computer, the armatu re windi ng res ista nce is usu all ~ t aken equ al t o zero to m ake Ilu x li nkages cons tant . Thi s .assum p~ion facilit a tes the solu tio n to th e prcblem , but leads to an inconsis tenc y of th e analysts and to a number of contrad icti ons. ' I A s hort-ci rcui t t rans ient pj occsa includ es several s tages . If a mechtne has a damp er -winding, t he first s tage is determined by a direct-ax is subt ran slent inducti ve reactance

'r aY

.

'

I

x d = x Cq -Ti ,,"':;C";'-;: ;C:7: 1IZqd+1 /%c/+" I!% cdd where X C q is th e lea kage i ndu cti ve reactance of the armature windin'g; ' Z a d js' t he direr t-a xisl in ducti ve reactance of the arma tu re; xal is t he leak age inductive teacta llce of the field winding; and z"dfl is th e dir ect-a xis leak age ind ucti ve reactan ce of t he da mper winding. The peak sh ort-ci rcuit cUlt en t here is

1, ,,, .0=

E m/;r~

where E m is t he peak phase -emf. The second s tago of the shqrt -ctrcutt tran sient s ta rts at the in stant when the 8rt-latur'a flu :!: pessee th rough t he damper winding and hegins to travers e the field wind ing. Th is condition of t he mach ine is 'defined by t he direct-ax is trall s'i ent rea ct ance

x' -x T~ . all t he cu rve of t he short--c ~rou it current we call s ingle out a perio dic and lin ape riodic component, The ape riodic component dies aWIlY wit h the ar mat ure tim e constant T a which depends on t ho armatu re indu cti ve reacta nce end ar ma t ure restste nce,

6.2, Th_ EqulIllolU of Sync hron o uJ M IIChl nu

l2 .

T he aperiodic com ponents of armature wi nding cur rents produce & fiel d th at is s t ationar y with respect t o. t be ar matu re, t herefore quadrature-ext e loops als o take part in th e t ransient process. Consideration Is th en given for t he t ransient and eub tra nstent in ductive rceccences alon g t he q ax is, Tbe g-ax is sub t ran si ent ind uc tive .reec ta nce is

~ _x q

o.

+ 1/ .1"0'1'+1V~o dq

where .t.'!' te t he g-ax is ar ma t ur e reactance; and leakage reactance of th e damper winaing. T he q-a" is transient reactance is z~ =

Z o"

+ Z ~ 'q

ZOdq

is th e q-axi !J

= z"

T he a pe ri od ic com ponent of t he ar mature c urrent oscillates lit do uble fre quency between t be currents E",/x~ and

It

e.,./x;

T hi s a nalysis of co m plex processes in a sync hrono us machine ca llsfor man y ass umptfous . Ne \'cr tbc less . H di sclos es well t ho phys ica~ pr ocesses and gives suf'Iie.lent fy acc urate resul ts. In the an alysis of the stat ic and d'yp a mic s tability of t he parll.ll el operation of ayn chro nous machines , wido use is m ad e of lin e incremental equa tions. The Incre ments of variables are ta ken l ine ar and t he t rea t men t Is g i ven to t be modes of small osc illat ions, T be s t udy of sta tic s ta b ility on t he basis: of s mall harmoni c perturba Lions is ju s tifia ble ein ce ill t he pt olJlems involved here account should be t .aken of the parameters of t he su pp ly line Slid electric machines and t ran sformers ope rated ihto t ho anme network together wit h t he s y nchronous me chtnc un d er :llnllly~ i 8, w The cr ea ti on of tu rb cgenerat c rs of n'IHli t power of 1.·2 to 1. 5 m in kW in the last years and the emergence of more comple x powe r systems ha ve ra ised new problems rel arli ng to the s t udy oJ t rens tents, in sy nc h ro nous machines. There is Il. need for II more rigorous en alysis of t ransien ts i n asyn chronous ccu dr u one, a t rt's t ar tiJlg of otterna to rs, d uri ng rou gh s ynchrouizatton ; lind in ot her emergency conditions of operati on of synchronous machines ill t he power sys tem s. As regards t he anal ysis of t rllilsientp in synchronous machi nes, of much interes t is t he illves LigllLioli of t prsiona l vibr-atlons of t urbogenor a tor s ha fts und er \'8ri 008 emerge ncy con d itions with cons tdernli on for t rans tents in the power s yste~. I n the ana lys is, t ho el ect roma gne ti c torque and t he momen t of ,i lJerli a are taken to he distribut ed fllolIg t he rotor leng th . The eq ua t ion of motion is t hen solved alm ultane ously wlth the vollll.ge equatio ns, A mos t jud icious a ppro ac h to Invea tigati ng t he d ynam ics of aynchronous machin es is to lise com puters for th o soluuon of equll li otl8-

ra

,

Ch. 6. Mutl iw lndinQ M-.:hl ".f

In con junct.lon with t rad itiqoall y adopted tec hn iq ues. The guideli ne for t he anal ysis of ulien t rPole and aonaal tent- pole ma chines wit h t hree rotor win dings an d t....;o stat or wind ings s hou ld envisage the U!8 of universal rout ines to ~e dll!llgned for so lving t he 8t8 ady -sta te a nd t ransie nt perform ance on digit al comp uter.l. h ill of much im.portauce lO t.e5t a nd make th e right choice of models obtai ned by t he meth od of experi men t al 'design. Th is wUl open u p new (Kmib lI n tes for st udyin g th e perf0f,mance of synchro nous machines.

ft.J. The Equ~Hons of Direct Current Machines Direct eurrmt

IJIllch int,

are mubtwindtng

rnachl lUlZ.

Most tdI Mtatle

d14fN1m1 of rtol tk motors lind generat or, I:on be trruu/orrMd to II , {napl l/u d modtlsuch as ilI listr at.kl in Fig. 6.2. In the model of Fig . 6.3, ;

d

-. -, q

...he a rma tu re windiog is sb;own t.o co nsis t of tw o symmetric win.I de gener at or t he 1I1!g-lIla r vcloc fty call be ce nstdered CO Ilstan L T his enables us t o d i.~reg i! rd t he equ a tion s of mot ion and limit the an alysis to the sBJutio n' of voltage eq ua tio ns. l it the ana lysis of t he d )' Hll m ic pcrfo r mauce of a machlne, it is of i mportan ce to determ ine correctly t he emf e, .. as a fu ncti on of cuerents an d to ta ke tntc account t he magneti1;ing and demag nf.'tizing forces. This can Il l.' do ne b)" use of t he 8- H cu rv e. t rnnsie ut resp onse, and relatto ns ((j.B)

wher e C. and em are th e coe ffidell t ~ that aceuun t (01' the cn eeecteessti c featu res of the mac hine ! under analys ts: an d (f) , ... is the result/l nt Il ux determined from tho transient respons e. Th e a na lys is or tr ansient ' ch ar acter jsuc s on an pnalng com puter req ui,-eg construc ti ng II cOffi .l{licflled ma th ernauical model or necessit ates t he si mula tion on a digit /ll co mpu ter. Al though t ile s tea d y-s"t/l t~ equa tio ns for l\ de ma chi ne ar e t he sim plest , t he stu d ies OIl t he d ynamic performance of an asy mmetric m achtne involve t he soluli qn of cu mberso me nouhuene equ ations . In Appe ndix I (see Figs. 1\:5 and A6) lice shown t he block diagram s of models for the sotu uon of equ at ions of de m achines. ..\ s mentioned earli er, dc ' mach ines diffe r from synchronous much i nos in Lha t t hey have cnm mutatcrs t o rect ify t he al te rnat.ing em f generat ed i ll ur e arma ture {\'i n dings . A mech e utce l freq uenc y (""O Ilvert er , or co mmut a tor. keeps n r igid t ie be tween th e anglll nr velo cit y and elect r ic frequency. while il semtcon duc tcr freq uency cneverlet may fl fford n flexible ti c, Despite th e fact t hat sync h ronous machin es have much in common wit h de machines, t he the ory of either of the t wo classes con t inued to develop se par ately for m a l~ y yea rs; com mutation p recesses were g iven treat ment in COHj U ll~ ti9 11 with operating p roces~es in 11 machi ne. I n consideri ng t he processes of energy COil vers ion in t he air gap of a dc rnachine, i t is q uit o jus~ jfiab le to employ phasor d ingr/lms and uquivalent circuits arte r red;uc iug t he multip hase ar ma tu re w in di ng to a two- phase win d ing . i n ~t s classica l deeign, t he de machine is II aalt ant- pole machi ne wit h a ;s lllt iollllry field winding. However , COD-

6.• . l ite Ooub le Squi ". I-C_

lnductioft M.oIo.

t roll ed- rec Ufiar com mut a lor mac hines widely use s ta tio nary ac w ludlu gs. N"o" stl li~lI t .pol e ue machines wilh a cn mpeastlt io, wind illg someti mes fi nd use i n practice. T he geuc r,Hzed 3pproa(:h to .studying sy nch ronous and de machi nes will promote (urUtcr the t helr ry of elect elc mach ines.

6.4. The Double Squirrel -C::age Induction Motor. The Effect 01 Eddy Currents WI! rest rtc t our IHUml ioll to t he eq uations or t he machine with o ne s t aLor Winding and two roto r wind ings . T he ma thema Lical modoJ or t his t ype is ap pli cable to t he ana lys is of I wido etsse of ml

I, I,

Cl

",

"

Cl

".

Cl

u,

>,

(C.26)

I.

'f

I n Eqa . (6.26), lind ere t he vol tag e nnd current matrices of thy lth machi ne . I n th e impedance ma trix of (6 .26), th e sq ueree donole t he im peda nce mat ri ces of 8 ma chi ne wiLh a circ ula r fie ld. Each im pedance mat-fi x eouse tna c9rrespondiog para meters. Theelect romag net ic torque here is eq ua l t o t he sum of prcd ucta currents in each e le me nta ry m achine

or

(6. 27}

To rque equa tio n (6 .27) for the ec mmon-r ot c r ma ch ine mod el i neludes pair.....tse prod ucts of currents ~ n t he It slators and the rotor a par t fr om the pr od ucts of eurre m st n eac h ele mc llta ry ma chine. I n t he common-r ot or m ode l , th e rot or establishes the li nk bel-ween n mac h ines. For the ser ies- con nected elementary machines opera ti ng in the sleady-s tll.te co nd itions , the Hoe \' ol ta~ is

v, = VI + V, +1... + U.

(6. 28)

and for the para llel-connected m achIn e! , the li ne curre nt is

i, = i + 1. + . . . + j" . . . . l

.

.

(6. 20)

where UI> V " .. , U" an d II ' I lo . . '. I . Are ure vo lttges ond euerenre i n ele me nt ary machines. If, apart. from the linkage between the clementlll}' pieces d ue to the ro t or cu rre nt, we co nsi der t he li n kage resu lting fr om t he ," ach ille

".

en. 6. ~u'ItJwl nd;ng M ...hlne,

Mturation. the impedance matrix in (6 .26) will be fill ed ec mplete ly:

,

:,1D O

D O

0 0

0 0

",

D O

0

0

0

9

".

,0

0

" (6.30)

• "

'.

The torq ue will then co nt~ in not Dil ly tilt> products of curren ts in t he s ta tor a nd ro to r bu t a lso tho products of c u r re ll t ll w it la d iffore nt sii08:

+ J/, + . . . +!lI,,+Mu

+ . .. + J il l + . . . + !HI" + .. . + M eA_u " (6.3\ ) Eq ulllions (6. 30) lind (6.31/ll ro l' i lfl i l~ r in s tructure to lb n equaue ne for an m-n wi ndi ng mac line. The ('quIIl io nl! for the gtlncrllli tcd energy conver ter permit the ~ llilol ysis of en ergy cea versic n processes in elec t ric machines with due rel:,lrd for man ufactu ring tncccr e. Mac hin i ng t he s tator nnd rctce ca ll also !lfreet t he chn racle ris liCJI o f lin energy converte r. This factor can be all owed for br consrderi og th o di fferences bet wee n lhe pemmetere of e lemo nta r y machtnea or l he p resence of edd y curre ut Ioops , as is done in s ees. 0 .4 a nd 6. 5. A set of var io us ma nufftcl\lf ing t ecrcre eonun c uly de t erm ine the per formll nce of II mach tue , a nd coustccrau cn fOf each factor i n t he se t mek es the an a lysis a very d if Hcull prob le m. As noted above. the equeuo ns for the 171·/1 win.ding mach ine permit t bQ study of most of these fa ctors , each ISC pa ra,le ly lind a Iew si multa neo usly i ll One com bination or a nother . Equations (6.26) a nd (6.27) a pply to th e ana1rsis o f pein t- wi nding machines i n whi ch t he field d i.s tr i but ion lit the ond portions a ro und t he periplltl-ry d iUors fro m that close to t he center of t he a ir ga p.

... 7

Chap ter

Models of Electric Machines with Nonlinear Parameters 1.1. The Analysis o f Electric Machines with N onlinear Parameters , As noted cllr li er, th e eq uauo ns of e lec t romechen tcn! e nc'l:Y co nvers io n with constant eccr tteteoie nm Inonli nea r equa tions IIInco t hey contain t he prod uc ts of va r- inhic s . Th~ l\ntlly t ica l so lut ions to 1I1ll~ eq uati o ns d n not exist if (0) , u ndergoes c hnngcs. Const oee th e "Hect of no nHnu r coeffielo nu i n tIle clect.romechantcs l eq ua r toes n n tho p rec esses ('If e nergy ee nvees toe in eree rete ma r h illes. nllmely . t he coeffici e nts L , M , 1.". r " r" lind J li nd i nde pendent "aria bles u , J. an ll 111, . A ll ccetneren ts en teei ng into t he eq ua t ions can be no nIi nen r. T i,e eeststeuce of II rot or ch an~s w tth cu rre nt displllceme Dt, a nd tha t of /I sta tor wit h helll . T he in d uctive renclll llCC de pe nd s o n M t unlt inn . T he mome nt of inor Lili ac me dr tves is II Cu net ion of the a ng nln r spee d . , T he pemm e tcr a dep e nd nn vo ll tl¥C9 . lond , and other- facto rs . vul i n ge ne rnl t hoy lire funcUon" of t ame 119 is e1enr fro m Bqs . (7.1). (7.2), lind (7. :1) :

ii,

.~ I:'::~~(I) :, (I) At

."

d

Iil M (t )

ra (t J+ d

I

0,

I

\ L 2 (t ) w,

+ 7I Lz(t, ]

x r z (l) +

o

Ill,

- L,. (I ) 10),

o

+ :, 7( t) I ,

I dT 1W;(l)

:, M(t)

(7. 1)

"

rl (t ) +

,

+ 71 Ld t )

lIf. _!If" ( t ) (IAli. - i~I ') dW, .'d t

'.

I

,

- N (t)

. . '.

o

= ( p l J (Ill 1~1f. - M , ( t )l

'; (7. 2)

(7.3)

T he modol of a ll e ne'1!:f co nvener. ll!l sho w n in Fill:. 7.1 , cc ereepe uds t o Eqs . (7. 1). (7.2 ), nod (7.3).

'"

Ch. 7. Mo de l. ot Elect, lc Ma ch ine .

-,-,.. • •

•

"I a

,.

• •

+ "'iii

7f "' h Ill

L.l a

•, , • •• '-"71 .""Tt"M:•.....

i- JotUQ

··

" -a-1>ltl " '1

---.. · . .,.· -,. ·"'. -- ·"'. • • • 0

o

-"

•

-" -:..

• '"•

."

:, )I', . .a.

n _

7s ot the te n ntJlals .....e ca ll rep resen t i ~ as a mac hine wit h const ant parameters l'iid noos in llsoidlll voh ages lit t ile in put . T he equa ti ons for II sa lllrtlted mach ine d iffer fro m those (or a n in-

1. S. A ne lyoi. 0 1 Ope,ol ion 0 1 Roe! El.. d .k Me ehl n..

'"

deer ton machinc in tha t t he forme r contslu coe fficients M~,,, , M;...,,", M;. ~ , . ... M ;:" ~. M~h, ... ., M~:~ ~ . and a lso other eocrne re nts to acc o un t. for the Ier ruma gn e t.ic coopttugs between harm on ics . Le t us la ke n look a t {JIO effect o r It nOlls illllsoid nl dte tr tbut tc n 01 tIlt.. ma gJll! ti zing Iorcn o n the s pec t rum of harmonics i n t he lIir gn p lind gi ve t he mathern u tical descri pt(o n of the processes umler t he assumptto n t hllt t he ma chine of m tc rcet is unsa t ura hlc flno th e vol tn ga is s tnuso tdal . If we ass ume lbnt the number of he r mcnics Pis infi nite nnd t he IIi.. gap ill smoot h ' th~ 11 th e a tr- gap fi l'ld repe a ls t he pnt t er u of dt strrbuuou of th e m~go()ti ~"lg force. Kn owing the spectr um of hllrm Ol1 ics i n t he ni r gl' p; nam e ly , t he ir emp jt t u dea a nd phases , we II p pl~' th e eq uatio ns for tho gone re fizcd energy conv er te r and set u p t he rna the matical model Of 1110 m ach ine t o duscrtbo the onorgy eonve rs rou procossos . T he aSSlimp tion IU;ll\! is t ha t the st a t or end ro t or enn-y t ho sa me num ber ofjiCU tiOliS wi ndi ngs, wh ich corres po nds t o t he c hosen number of h .r mnn ies . T ile prob lem being sl a ted rnua t cover two to Jnu r hermouics. Im d it s solutin n ca n not corta inly be eccum te. T ho eestereuces o f fi ct itio us wfnd lngs IUII Y be t uke u eq ual j.o.Ihe restet nncee of ect ua l windings. 'I'he m u t ua l ind uct an ces associ rted w ilh upper hnrmo ntcs of th e m ngn llt izillg" Iorrc ma y be tak o rr a pproxima tely e q ua l to onet,hir d of t ho Ion da mc nta ! for t hc th ir d, lJarm on ic , to o ne-I ift. h for t he Ii fth he rmont c , etc. T ho coofft cten ta df coup l ing between harm oni cs, .U n /. c,011 001 be h iqllor th nn t ho mUllln l lnduot an ces between upper ha rm on ics , ,11/1. T he coeff icients Oljle riu l;l in t o elec tr omecha uical eq un t ions de pend o n load . T bus t he eq uati ons for tile ma ch ino wit h o »oestn usotdat mmf di st l·i bu\.jon arc ~ h lJ same R~ those for a satureted m ach ine . Th e oq ue t tons differ from each ot her hr the values of coeffi ci o ut.s ami the om p l it udcs of lutr moni cs . For II nOIlS/lt llJ" lIted m a chine , th e 'I, m f .liSlr ib ll t io tl ill ainusci dal . the "ir ga p is smc our, th ou gh oOIll,.(uifor m dill! to man ufact uring fac t ors ( m isa li~lImo ot of the rotor wi:lh respe ct \ 0 1 11l~ SIOt.Or. elflpt icity , con tc tt y . etc.) . T he ate-g a p numnuro rmt t v is rcs pouaihle for th e a p peara nce of t he s pec frum of ha rmoni cs in the air gap. If t he ga p Is no nuniform botl l in t~lC axta l find in th e rad ial di · rccuon . for li ll a lpi ng the processes tho mac htne is br oke n in to III p ieces in tlt e axia l d ire cti on. and t heimachine mode l is cons tr uc ted with In res i! t ill g torq ues a lo ng W (' :T. y, a nd :: ax ea. As we di d for t ile mll('hine wilh two degrees Creed om , he re .....e lloN I t o inl rol'luro o nc m e ee eq untr ou i ll i)rllt'r l iia l t ile 8ys l em of llil"il l ions for t he m nch iuo mut e r ~ ~ lId y sh nll1d be det er mjue te:

.>t.

or

w,x

+

w .~

+ 00,. =

w

Th us. tw el r;e equtd loWl, namely , tight /,Joll agl' equuliolls , three

(9 .8) eq U{l~

li ons oj motion , (lnd one "elocUy eqll!1.tion deu nbf< lil t. procesus of elleTgy renverdon /'1 a IIweh ine wf l h. t hree dt /lrL't1

01 ! rud/J/Il .

Por a sy m met r ic nHt ch ine li t J/, x =- ,1I., = .11., ... ,11r' Eqe , (9 ,5) t hro u2h (9 .i) become si m pler. ' Vbon .11u = .11c, = "ll . . ..nd J •• = _ J " = J " . t he vel ocit ies al anw t he a sp s a re eq ual to (lU I)

Mll.ch iue! wit h II. s pher ical rot o r Hnd oppli ca tio n in Illw ig8 t ion de vices . l f one of the 5tn lor" is mod e \0 re vol ve flOou t t he ro lor , t he m achi ue 10 des ig ned shows fou r deg rees " f Ireed orn. If two sta to rs eevclvc in d p.p6nd crl tJ ~' nbout. t ho roto r , th e mnc hi rw wiIJ ha ve tr ve dl'groCS of Ireedom . If \\ill r ig idl y CIH m OI: t t Wll s t a t ors II 1HI 1I 110w t holll t o revnlve ab ou t t he f ph el'i a nd h und red s of k ilo me ters a nd the ro lor is n (IIr slls pc nd lld a beve tim Hue bell . I n des ig lli ni mag netic-eus hi cn Irllllliport ve htc tes . the elillincc r lia.~ to eotve t he pro blems of cont rol lind sl ob ili za.l illn (Iev ita lion) of u peseeugcr c er, le t elo ne the problem of de crensl ng the cost " f gucl. Do t ra ns por l systarn. III t he a rea or linear mot ors t be re lire ye l ma ny co m plcx prnbJems t!la t a wa il Ihe ir solution , of which the DlOSL co mp lex o ne CO IO\lS 10 t ile fo llow ing . A5 far bec k U tbe m iddl e 1930:1 electeie ClI lop" lt.s were bu il t wit " l he e lm t o im pa rl a ll I'I dd ilional e ece te eauc» 10 fl)'in: vehicl es (F ig. 9 .5). While i n 19round l nll rnlport aj-stema t he ga p be tween t he bed a nd t he cor must be kep t accura te t o II h igh degree . in Cll in pu iting Ihe {l'ap is mad e 10 vory (t he rot or m e s into space lind t" o per e me te rs i ll equeuone und ergo clu. ng{,!I) . I n IIID la t ter CIIM! t be re is 8 nerd for ca tc ulnt'ing t he d r iv ing terce li nd RCCe-lerat.iou . Mot e d iffic ull pro ble ms Mise III lin nuo m pt t o brl "g the rolor he ck a nd tak e off 1.1'13 defi nito a mounts of t'nerg}' from h to ('ffeel the des ired decelerlliion . Alt houg h they lire not devoi d of s horlco m in{l's, line a r mo la rs e njoy n!le in graph plo llers. mn nipllilito rs o f me lAl p tecee. pus her" , and I" orhe e etecrne d elves . Il e\"ersi ng the. motion of a li near mOWi r g ives

Ch. 9,

~qu"'tl on.

[0'

~ [ ,,( !,it

Moch lne .

a n osc ttle tcrv-monon mot or. T he ana ly sis of Hnea r mot ors ell"lbJes us 10 e xte nd the result s t o rtet er mt ne th e rcleuon between electr ic machines nnd appara tu s in wh ich t he d r ivi ng el cm ents mni ni y exec. ute Iiuear dlsplae omonts w ith v ~ r~' i ng pa ramet ers of ctecu tc circuit s .

9.3. Energy Converter s with Liqu id and G aseou5 Roto rs TIll' st a tor of II li UNlr mo tor can be built in t he for m of II. pipe Ins ula of whic h a t raveli ng fillid ca n 'b e set lip. I n t he pipe fill ed w it h n magneti c Iiquirl or ~1' S f~ mov in g conduc tur}, l h\! mng ue tlc

,

)

,

Fig. 9.6, An ]'H! D generator

fi eld \I'm influe nce t he mot ion of th e flui d . If An iouree d gus I'll plasmu) or il lllag ne t ic li q uid i1; d r jven throu gh t he chan ncl , we ob ta in A ll energ y conver ter called II 'TllagJlo lohyllr ody,mmic (MHO) generato r (F ig. lUi ). An MH O generator conv er ts the rnecha nimli (ki netic) e nergy of pbsmi\ par ticles t o etcet ric tcne rgy M t ho conducting plasma Ilcws t h re ugh a ch an nel l III wh ich magne t fc co ils 2 pln ced nlougsid e t he p ipe pr nduco II magn eti c fi eld IJ . Th e cond uc t ivi ty of the hot gas grow9 with the ad d iti on of 11 11 easil y ion ized ~!Sull.s in t he are a of magnetic-field ECs i nto rbe area o f elect ric- Hel d EC& a nd t he n. Cive n t he ID l'l th em at lCBI d GScriplio n of the processes of lIll(lfn' conversio n in electr ic-field electrome cha nica l sys temS. to design t he converters of t he dNl lred performance. . Th e eq uat ions d8.\lCri hi ng I t he processes of e nerlU' con version In mag netic-fiel d nod alec tr te-Held el ect rcmeeha nlcal s ystems rerollin t he sa me e nee the interchange of du al pairs in t ho follow ing tab le Elact d e cbarge Q. Electri c nux 1lI. Vo l l.a ~ "

-

- - - MagneLlc charge Q", _ _ Magnetic /l ux tIl",1I _ - _ Cur \'('n ~ I

Eleetl"O!IKllive lo('Ce .

-

- - - ilh gnet OCllo tlvt: force ...

Th e ad vantage of t he eeoee pr of magn e Lk ch arge is t hat we ca n est.ablish the d ual of th e equations for a n electrom agnetie field. T he mat hematical doser ip tion of t he phonomena i n elect ri c fiel d! t bm ~'ie lds the beh l v lor i!jl magne tic field s. Sued on t he t boory of;duel . irwerse S)'st.ems, we wri te down t he foll owing up~io n.s r =0

_

(dQ..l dt) ... - (dV../dt ) "'" _ (d ld t) ~ S;dS

- -I (.1J"t)a~+ 1(ii XB)aJ

,

(10.2)

h= - (dQ. /dt) .. - (d'l'.Id l ) >- - (dIem ~ DdS

= -

J

(1)1)10/ ) q$ +

I (5 xv )

HlvJ.. B J.. dT au d uJ.. DJ.,. dT,

dl

from (10 .2) and (1.0.3) we r == Blv h -"'" D 1»

(i O.3)

,,, (10.') ( to.5)

where l is rue le/lglh of I cond uc to r in a ma(lletic-fi eld EC t hnt ill equal 10 rue wid t h of an e lectrode in I n elC(:lrie-fit"ld EC. Th e phen omenon of elel tromagnetic ind uctio n ts put to use in mag neti c-f ield ECs. Ind lilt of etee t resta ue i nductio n in etec wtefie ld ECs. S im il a rl )·. usip\:: two-phase , t hree- pb ase , al'id m.·p hl!lll s yste nlS of e lectrodes , we ~n produce rola ti rli electric fie lds .

lS.

Th e t o ta I en erg y of an e le ct ro m egnetjc Held is gi,-cn by

W _ lY.. + W.,, = 0.5' ~

(ED+/iB) du

(10. 6)

•

Here IV.. = 0 .5 ~ ED dl.' is th e ellergy of a n electr tc field; /Iud W ",

=

.

0.5

JiiTJdv

i s tho ell\"!rgr of II mngnct ic fiel d .

• p rocesses

T he of {J.n(' ri:'~· con versio n in Jllllg lle li r.·fie id el ectrome chan ica l sy s te ms rea u l t fr om t he i nter net.to n of magn eti c cha rges Imaqnc t te poles) a nd build u p of an elecrnc fie ld whose S O.lI t rE' S a re electric ch a rges . E nergy co n versio n In electr tc-He td sys te ms s te ms from th e Inte r-a ct.ion of ete cer tc cha rges lind bui ldup of a magn etic field wh ose sou rces lit e magn e t ic ch arges . T he equa t.ions tha t can adva ma geously for m t he bas i.!i of t he t heor y of elec t r ic- f ield ECs are t he sy s te m of e q ua t io ns which a re d ualinve rse wuh res pec t t o Ma xwell's equa uons for movlng medi a . T he theor y of t h ts c l as.~ of E Cs nl fl y re ly nn t he e qua t ions a nalogous to those for mn gne t.ic- ficl d E Cs .

10.1. The Equations for Elec1ric·Field Energy Converters P r oceed i ng from the t heor y of d uet-r everse el ect eodvne m tca, we form ulate t he fo llowing d ual- Inv erse e q ua t ions for th e generali eed e lectric-fi e ld energy conve r ter \ls i~ng Eqs. ('1.34) a nd (1. 35): i~

t;'

"il

g:' + (d ld t ) C:' (d /dt ) C

(dJdt ) C O

g~

+ (d/dt) C;.

-Coo,.

- C~ oo ,

o

0

CID,. g~ + ( dl d ~ ) C~ (d /dt )C {d ldt)C d +(dldt ) q

X

('I Me =C (u~;' - u :,u&)

. f U~

O

qoo,.

"I "I .7)

(10.8)

Equat io ns (10 .7) and (10.8) follo ,v from ( 1 . ~) and (1 .35) lifter t he in terch ange of u a nti t , tnd oewnces L:',~ a nd tOl lll onpncttances G.'.'n . mu tu al Inducta nces M lind tn teroleetrode capacita nces C, e nd r esist a nces r:': ,'6 find con duct a nces .G~ti.

18 2

Ch. 10.

e l u j rle - F i e(~

end Eledromlg ne tie-Fle ld EC

T he tota l ea paeil/l.llce'jinCl udes the capacita nce C bet ween ure sta tor and rotor e lectrodes:a nd self-eapacitn nce 4:: C~ = C + c~ (10.9) The dua l of t he ge lle ra lite~ jnagnetfc-Ifeld energy converter is t he generaltzed elect ric-field EO with elec t rodes a t pote ntta la u:.:.'a instead of windi ngs IO:':~ . electri~/ield ECs , ltke magnetic-] ield ECs, indude syn chronous, asynchronous and , commutator m(l("!linl's, and a~o trans formers. I " II sy nchro nous electric-fie ld machin e t he vel ocity eo, = w., a nd the rotor ta kes c ur rent from 9 de circuit (Fig. 10 . 1). Elec t rodes A . B, and C (which replace wi ndings) prod uce a rot at i ng ele ctric fie ld , a nd inte relectro de FIg, tn.t . A eYllcbrooQus etecmc- capacn a nce a nd self-eapaetta nen prese nt in t he machine vary in a fiel d machlnl> ha rmonic manne r with field tot ntion.

T he d irect- lind quadeature-ax ts eq uatio n for a s ynchro nous machines has th e form

g~

=

+ (dl dt) q (dld t ) C - Cw,

o

(dld l) C OO u~ (dldl) C~ C;()l, cu, u~ - C~ W r g~ +(d/dt)q (d ld t)C X u~ 0 (d ldl) C g; (d{dt ) C; u~ (10. 10)

g~ +

+

M . = C[u ~ uJ- u~Il ~ 1

( 10.11)

Here u~ , %, u~, uq are the d.g stator e nd rotor vol tages l n t he t wophase mac hine; i~, i; , I~, ~ lire t he cur rents in t he sta tor a nd rot or electrodes: n, g~ , g:i, g~ are .the cond uctances of th e ste toc a nd ro tor elect rodes; CJ:; represents tota l capecttnncea equid to a ~~ =c+,~~

where c;',~ re presen ts th e se lf-ell. paci~ nces 0 11 t he sta t or a nd ro tor elH'trodes ; a nd C s la nd! for ea pac itll ncell be t ween the sta t e r a nd rolo r e lec t rodes . An asy nchro nous electric-field meeh tne comes from t he sy nehro-DOW! mech t re if i n t he la ller we repla ce the roto r by III dielect ric d isk or use a rol.or th at I. u the sa tne n umbe r of phe eee as t he s ta to r .00 a pply to the ro lor II volt age at a sl ip fre q uen cy . A commuta tor maehilUl CA n a b o be bll il t lro m t he s y nchro nous IDlc,hiDe by i nser ti ng II co mmut a to r lnt o t ts lie d rc uit. In a n elecrrt c-Ital d t ra nsfor mer it is 'H I el ec tric fiel d t hat li nk. the e lectrodes. T ho eq ua t ions of this 'tra nsfor mer have t he form I,

- It

I=

I 1"'1 u,

1 8 1+

(d ld t ) C 1 (dldt) C (d ld t ) C 8t + (d ld t ) C, X

(10 .12)

Hero subscripts 1 and 2 s tand for- ~he pri mary lind secondary respeet rvel y. T ile curre nt t rllTl. ll11tll'oetlC lipId e nergy in t he nir I:fl p, e tectric-fiald ECs must co nce uwe te the elt'ct r ie-fio Jd e net gy in li qui d or sol id dfelee tr tcs . SlJlOO t'nergy conver ters m ust have cle ments movi ng wi t h res pect 1.6 QUit a not her , eleetr-ic-Hcld EC design.'! prov ide for mecheniea l c:aps or ellVis.'\go llle use of liq u id me teete ts to ~T\'C t he p urpos e of a rotor ,

or

Porametr tc tllclrfr-!Uld t ntrlU eolw~rtul openue on I he pr lndple

01 a ptriodlc ~~ ngt tn c:apa~ltll~ at t Onst4llt u c:ftation: I;olta~ U•• All exam ple of I llch a n ene rgy co nverte r is II gcn etRtor uti liz i ng t be onergy of SlI rf (f ig. 10.3). Milny a tte mpts to co nve r t the (lnerl )' of chaot ic moti on of 5C1I waves to el! hl' tho test clHlm bcl' tem perat ure X " lll/lchilll' vibra ti on Xz (of cour se more inte n!'ivl:! t ha n t he n",r1l1 11.1) , e ngula r vclccif y %'~ lind t ime (Bc t",r t wh ich is c.erta ird)' peese ur in t he ex per tmc nts of tllill type . T he /'t'b Uo " : > k" = f 1:%'" :%'•• I) can IK' found by IL~ of t he fonn nli red EO peocedu res .... h ich ill,·ul>·e t he fo lle .... illl:: aa t ller inf; II prillr; illfor mAt io n: dr term iu illg th e cu rvlll un! o f l hl' fael or space llt roul;:"h tbe c:mll!~rlI et io n 01 cm e-d hnt'll s illllill eceuous of ""ft = f (:%'1) ' k" == - f (.I'z)' il llt! k " - f (.z-, ): d tfJOS illg l l~ lowur d lld up per linl ll.! 01 va,i, Lion of Ihe Is etc rs: carry ing "" I t l ~ I'l:pcri me llt on a p propr illhl les t tees: gi " lug t l.... mM hemal it'.1 l rt!nt men t of t lo" rcanh a lind Inter pr\ll ill8" t he-n . Suc h I't way of the d it"('cL s ulutln n to tho s ta te d pr ulilem by ~ ll tl fOl'lllnJizl'fl ED procedures meall" t hat we havll to realize t111l matr ix: of n po wer IIr at 1f~ 1I .~ t 2 4 (Ta b le ll . ;~l' T he ED t echnlq no erHlhleli liS to perf or m t ho tes LIl ~ uc~ss rl1 l1 ~' 11 11 de te rnuue til t' rci llti ollS kft _ - f ("" ;tz· %' z. tJ U t he co ust rsl nts 0 11 the sc o pe. of the ex periment (i.e . the number of mad linI'll 10 IKl p ill 10 test ) a rc rensonahle And 11 0 a mp le a mou nt of a pr ior i tnr erm e uc » i.'!l evelle ble , I1 n m e l ~· . l loe res ul ts of pre li tn iIlRr }' er per trnent s lind Pro ll,erl)' chose n vllrh. t io n U II ~S , parti cu lar l)" th e var ia ti on ra nge of t he t ime Ieet c e. Experi l" \c:e shew! t ha t if is more a.d'·ll nt allOO".'!l to 1L."Cl not th e stAnda rd E D rne tbods I". t a some ....h.. ~ refined t« h lliq uc bll ~d on t he se mo r ••lell of t ill' E O l hen ry , Accou nt must be t ake n o f t he (ael Lhl'l Lan incrt'II!8 in t he scope ct t he experi me nt requ ires I'l corf('!Jpondi ng Inc rease i ll tile n umbe r 0' test beds a nd in the t est t ime . I II so lv ing the reli a bil ity prob lems a nd es tlm nting t he fUlieli onAI re lat- io ns of t he t ype k .. --! (.t ,). n li tt le-k nown bra nch of olect ric mach ine l'ng illet> r ing hal! 10 be kept' in mmd. wha t ts meant he re is the physi cs of l hl:! pre cesses of n g ~Jlg a mi wenr of elect r ic mnchlne elem en ts sud . II ~ comm uta to rs , bellr ings , sli p r illg3, and II win d ing un der t he cnmplex Ilcl ion of ele" Ilte d tem pera t ures . int e ns ified ,' 1brnlion s. a nrt increased s peeds. T ho d a ta on t hese precesses , Il;l Lal ene l he da ta ill a nnJ)'liclll Iorm , is usua ll)' not Ilvllitoble for a par li cubr mllch illoll or for II loall'h of t lK' mlld ..; n!?sof Lhe s.me ser ies. I n '\'i ew of t h ill file t , setting lip t he "ll r i... Lio n limits o n th e ti me fact or bceo rlle:5 pr oblema t ic . An u nllerestimnl ed. " p PE' r l imit does not Illlo .... for th e

r.,.

T&!:> t. 11.3

Naeblne No.

~. t

No. "' I i ll

I ""

1 "',

I ..

1

a a

, s e 1

, 8

"

a a

, s

• •a """ "" "" " 7

13

+

... ......

...

+ + + + +

e 0

+ +

+ + + +

0 0

+

+ + + +

0 0

, ,.

"" "" "" ,,",..... ,,.... ,

.,

+

+

...

-

R_

TOUt t _ ....

+

+ + + + + + +0

.....

0

co m plete IltiliUlion o r t ile ex pe r ime n t. Th e o btal uetl values of the aceele ral.ed te!'t coe ffi cie nts prcvelcwar thnn t he possi ble oues , which , In t he fin a l ana tvsis , longt he ns out th e acce lera ted reliabfli t y t('SI8. A n () v t! ~s IiOJllted upper linl,it of th e lime teeter ma y render the ex pe r ime n t u ns uccessfu l because a t lea st oue un reeliaa hle row of the Illllt r ix mea ns It fa il u re of th e en til't! ex pe rfme nt or. nt best . t he transi t io n to an up per le ve l of fra ction i.'I&:. ie, th e l.n ll8i t ion rro m the 2· com plete fll(.torial to th e 2'--1 fra ct iona l fa eUlr ill1. from tIlt! 2· - ' te the 2"- 1 fa ctor ia l , etc. T ho se que nce of the so l ut ion 10 t h is pr oblem is a s fo ll ows . Prior to co nd uct i ng t ile ex per tme n t, t he levels of th e t imo te eter lire left normal. T he l ower , zero, a nd uppe r lev els lire round on ly for the acce ler a ted tes t l ect ors s uch as tem pera t u re , vfbrnt.ic n, end s peed . I n do ing th is , it proves possi blo t o im p leme nt the 24 de st g u rnatr-ix ror the co m plote factor ial nx per tme nt , T he fi rst Ie ctcr is t he fac tor of t.ime , T he time rec to r pe rm its n roni ng tl"sts 1I0t on Hi bu r on 8 ma cbiues in tlte 2 6 CF E . O ne a nd th e sa me machi ne ill pu t t o test bo t h al l im lower and a t t he u ppe r level o f th e t ime (a ct or .

20' Each of the t.es ts is ru n u nt i ! 011 the elamems und e r invl'Sl_ign l io n.

[Ail 10 operate . This mea ns t bllt if, for ex a m p le , a hoar i.ng sub ess nmbIy fails . the time of failure is recorde d . t he subnasem bly is replaced a nd t he ex perimen t is contin ncd , bot t he new coun te r-pert is tn ka p ou t of co ns id era tio n . If. Iurf.her-, t he s parking III the com muta tor-

exceeds t ho perm iss ib le leve l lind it is d iffi cu lt t o rem ed y the def ect , the c om m u t a t o r is left to s ta nd a nd the m a chi ne to r u n s i nce the

operator has rece iv ed the in format ion 011 th o tor lIud exclude d i t from c o ns idera t io n.

rcuure of

th o

CO m m ll l ll~

T he measurem en ts of t ile op t tm taatfo n param et er [cc nt r nlla h le par ameter a suc h a ll no rse, I UnOIl! lime , s pa rklng, Ineula t tcn res tslance) rue t aken cont inuo usly , whene ver reos tble. ra the r thau lit ind iv id ua l po int s cor res pond i ng t o t ho vctt.ices of t hc h y percubeund er s t ud y. This cnn he th e ca se for , ~ II~' , t he com m ut ato r unit e nd s l ip r ings . Th o a bo ve mea s ure men ts ca n a lso be mndo dise retuly f Of , snv . bellr in gs nnd insula tion over ins ignif icant t.ime Iutu r vuls , from 24 to 48 hours . Th is se q uence 01 rea li za t io n 01 t he desi gn mat rlx shows irupor tMlt features which m ake t I n'! solu t io n of I he stn terl pruhlem posslb to . Tho po t»t is t hllt tha mat hem ati ca l mode l of t he pr OCl'5S, t.e . t hepolyn nminl de pe nden ces of th e cpt tm taa u c n pnmmetce on th e t est raet ors. incl udin g lim ti me Iactor , pr oves op ti mll l s ince t he re cord of t he levels allJ u,« in ter va la 01 va r ia li on or t he t ime fa ctor is ke p t , afte r obt n in illg t he exp erf me ute l dnla lind pcrformt ng the ma t heum t-, Ical t eeat.men t . T he nex t ste p tn vcl ves tho ceu mo uon of t he accal era ted ll:!st teet ers fro m t he obtai ne d pcl yno m iul de pendences . A no the r a d vnntage derived from the ab se nce or fi:l"ed levels an d tut cr ve ls of va rsnt.inn i ll ti me is t he foll .o wing . If th o obt ul nad llill t hemutica l model i ~ inadcq u nte , r.e . th e m at hematica l descr ip t ion doesDot ccrres po url t o HIt! ron l process, we r a n pnss to the no ulinca r- t rnns-. for mll!,iofl of coor dlnn tes, replacing lor t l>tl p urpoi"ll t ile in de pe nde nt vari ablE'S hy fl ew o nes , " = ;r.~1 or ~, '= In ,( /> a nd t hus cc nv ert tng to the logar it hm ic or ex pone ntl n l t ime scal e . This fea tu r e does m uch, towa rd t he SU of Il n ca lc u la te d with t n ll pol ynom ial. T he di scr ep an c ies at s ome poi nts, es pac ia lly [or tosUl Nos . 7 and 8 a re so large tbet, th ey go 'beyo nd the s peci fie r! limits. A d iffer en t a ccuracy of pr ediction in va rio us dtrectt nns 01 t he fact or s pace , t .e. a t va r ious com b i na t io n of K u a nd K I, te nne of ti l\< grave disad vu n t ag es o f the seco nd -or der orthogona l desi gn . Suc h a poly nom ia l is imp ra cti cable Iceuse in . the ann lys is a nd sy nt hesis of the mach i ne becaus e tIle sea rc h fo.·,the optima l. geom ctric pa ra meters of the machine is dono for venous eooib rneuons ot K u nnd K I a nd t he pol y nomial e rror m ay.result in a fal se ex t remu m . Where t he a ccure ev of t he pol yno m ta l over t ho s pec ifi ed intervals of variation of t he fa ctors is of prinllll'y Im'P0rUinc,e , the inv estlgauoo of an al ogs a nd ph ysi cal. m odels shou ld rely o n ro t a ta ble ce nt ral eemposjte- d es ign (RCCD). T h is t yp e 01 des ign brings ab out an t ns ig ni.fic a n t in cre ase in th e number of t est s t o be porf orm od in comparison with t ho or t hogo na l t ype but offe rs a h ighe r acc ur ac y {T uhi e i 1.6). Yeble 11.6 Tho Numb ' . of TeslS R1Ill In aC CD _ _ __ C... : : Rc e D' Met ho~ i de r t hl'l s t11leme nt or t he pro blem invol vin g ure ss timnt lon of a n npp roxi mill ing pol ynom ial for l ire speedup ti me t u ' of an inductlo u motu r w it h a 20 % dav iat.in n of nrc v etuo s of th e Icjlowtng qu anti tie s from th e nomi na l v al ues : rotor res ist an ce J: l • sta t or rest sla nce 7 2, II IH I momen t of lnor tla , .T~ , T he mnl ri x of the CFE. 2' deaign is ~i "I'l Tl i n Tnhl e H,i . U n der t he exper-iment is mea nt here t he solu t ion of d rtreeeuua l eq untto ns (:' .1) a nd (5 ,2) on a lJil{ita l com pul er n t th e Itxed va lues of (110 machine parameters in sec orda nce with t ho des ign mntt-ix . Determ ine t he pol ynom ial coefficie nt s from corre sponding des ign formu las , Since the repe at cal culn t tnns of the machine rnet ors at. t ile SlI lIle va lues of paramet ers gi ve the snme resul ts , th e ex per-iment var tanco is o' {I. p } = O. an d. hence, a l l coefficien ts of the poly nomial are s ign ificllnt. A check of, the model for ndeque cy agllimlt t he F- tes t ill vcr t tces of th e Iee tc r space can not be cer eted ou t. In snlv tng th o e pproxt mntlon p ecblc ms. t ho check of pofyuomia l ade quacy fOI· l . ,> is mad e b)' selecting- othe r poi nts in t he faclor space .

11.2. f he r echn lq",.. of fo p e,;", e nt.,1 D.,;g n f ebl. 11.7

Vod al", olll,..e'po,. ·

1~1 g'"l, 1,,0\1') .1

Jl ~e,ern:. Ul' p•• ln .\ncl~\ 1 + Lo.....

le... l

_

J' ..~1 8

3 . 08

"

13. ~71 •. 28 2.116

..

" . {Il,

h I m " 1 . &2 X 1 O ~ 0 1.82x lO~0

,

+ + +

Test No.

a

+

, s

G

"I'"

"''''

"

+

+ +

+ +

+ + + + + +

Ol>J.... uve

t unetl on

1. 22 >:10-0

.

1

2 3

" In

+ +

~.

+ + +

+

+

+

+

+

+

~

+

+

,

ro 4.~

3.'

,

II

e.s

+ +

.;

As is k now n . i n per form ing the ex peri me nt s fit ver t ices (Fig. 11.5) . t he Illfges t d iscrepa ncy be twee n t il> a nd t op occurs n t poi nts 1, 2 , 3 . 4. a nd lit c~lIl trlll point a. T heref ore , th e chec k for t he model a deq ua cy sho uld be done at these pctuts using Cochra n's test on t he homoa gene ity of inadeq uacy var ian ce

"

ast tm a tes:

,

.-.

o,na>: = max (S; .aV L: S~ "d S~

I

0

( 11.25)

a

'

.

ad {l op} = (t ,p q ill t ile inn rlequacy VDna nca for t he 'lth check po int; lind k is th e num ber of chec k points. ~ig. 11.:;. The cha rt for perf . I I.

Appl i t~li o "

of " , pllri me,,'.1 De . ill"

l'e-nt resul ts . T he Inost spec if ic eeses TOc t with in o p l im i~/I t io n of e le.:l r ic lillichtnes resu lt from II sha r pl>- red uced rl'~i oll of SHll'I: h . T hE' l onrli in whi ch lh ~ o bject i" e Iu netio n sa ti s.Iil!s Ihe imposed eons rre lnta a rt! smnll . so it is d ifficul t, 1.0 o bta lu s uch to nus for t he orga nization of Ilrpwi..o procedures without constroc ti ng I],Cl secuone. T he solutiun of opU mlzn t lo ll pro h l elll ~ of Ih is elMI!:I prose ut s d ifficu lti es IISN)Cilll ed wi t h t he cho ice of t he re rerenec point »ccded t o inlt la te th e di roc tod S/J llrc!, . T hi ~ is e tsc l,he CI\!IC for 11 lorge cla n of pre blom s tn volved i ll t he a na ly s ls of llle ctr lC mnehtnea. Wlu'll ncccu nts [or th is f/lct is t lllll a n ind ucno u machine liS t he ob jec t un dor n n(l lr~ is is II fllir ly wct t et ud led objec t lrom bnt h t he l \;eol~ U r'lI l s ud ti le pra ct ica l am ndpoi nt. T he mnch i nc, In se r ies pr od l(ctioll /Ire ro ~hcr etose t o o pl illlni o oos iJ II", Imp osed co nst rain ts are tak e n into accou m . H e nce. thu poss ibiUlil.'s 01 impro \'ing II.('ir CMl'llct llristks lire noL large., ,, 1111 so t bo ec ne or pcrralaaible sea rch for t he so lu t ion is small . The pol ynomial re ln l ionll a ud op ti mum sea rch ZOIl('S obla i llc{1 h)' th e ED technique u sist in choosing th e rt>fcre ncn poi nt needed for \I,e ol'lrllni l.o'ltion of the directed search. Problems lire me l w it h In pra.c ti ce in which the sellrch r,,:::-iOIl is d lsco aun ucus, III hllutlling t hese proble ms b~' nu meri ea.l Dl(l th od ~ , t l,en) Is a ri sk of m i" lJIki nJ: I lle l)Oint of to pa rt ial ex t ramum for the op t imu m peinl . The 1I 1ln lys ls I l_~ l l a

210

Ct>. 12. Syn lt>"'i, .. I Eled,i" ,....,ct>in• •

or t wo-di meus te na l sect ion zones helps t he engineer avotd th is mistake. Appo udi x IV prese nts t he ass tgnmo nt fOJ' the solu tio n of desig n opttmtam.Icn probl ems fO f an induction ma chine wit h t he a id of ED tech nique . App lying t his 'techniq ue , t he anal yst can tra nsform the gene ralieed m achlne m odel eq uauons i nto s imple polyno mi al rcteuons between t he c pera t.lng factors of II mach ine a nd its parameters . Th e ED tochnlque al se «ucwe ono to s ingle out t he ma in a nd secondar y foctors tha t affect certain ope ra ti ng cnerecteetsues. By use of t he ED technique . it beco mes possiblo to co nst ruct hi gherorder mn t hema tica l mode ls of enl.' rgy conver ters a nd pass \.0 geemutrfc progrnmmi ng .

Chap ter

12

Synthesis of Ele ctric Ma chines 11.1. Optimizat lo.n of Energy Converters . Optimization M ethods The objective of t he the or-y of elect r ic maeh tn erv is to lind t ho wnvs Ior the evo lutio n of now ellergy conver ters a nd o pt tm taa rton of cc nven uo na l t yp ClS. T he des ign of II ma chi ne Inolud ea th e stages or computa tio n lind enginee ri ng development . I n the gener nl case, eompu rat ton repre sen ts :I mat hemntlcafly indefin ite pr oblem wlt h many sot uuo ns beca use the number of un k nowns ill the problem is greater tha n the number of oquations. Th e wny out Is -t herefore to prese t the va lues of cer tain quantiuies reaso ning from t.he exp erience gllined in eteernc ma chino engineer ing . Of a few desig n versions, the most adva ntage ous. optimum one . is t hen chose n . A cr tter-lcn of' optim iza t io n ca n be 3 mi nimum total cos t required for t he productio n lind cpe rat.ion of a ma chtne, a minlmum mnss and toSl of tho ma ch ine, fi nd e ther- factors . An opt imi u t ion cr ttorte n depends on th e field of a pplitnt ion of II particular me chfne. t. ime, cos ts of materials a nd e eergy, etc. The se fa ctors complicate t he task of seoking a ll opt imum des ign . T here is a grea t num ber of methods of elec tri c ma chine deaig-n, Ea ch me t hod re lies on cor ta m mathemettcaf mode ls interrelating inpu t an d out put chere oterfstics of a machlne . Most me th ods usa des ign e que t.ions obtai ned both a na lyticall y a nd empir ica lly. Scmo methods employ equa cions deri ved from equi....a lent ctr cun models. In t he Iast yea rs a t tem pts hove been mad e to use d ifferent illl e quatio ns to Icem a mnth ema tica ! mode l for des igning energy co nve rt era,

12.1. Opl l",lu I;Of1 01 EC.

Oel;mj~"tion

M.n.oa.

The use oj dl//rrentilll equoUon.f 01 tlrdromn:Jl(lmcal energ!l eon ver'ion fo r dulgn purfKJU~ olfers 0 means 01 conllder/ nl! bofh :Il(d ff: un d

dynamic chQrtJCt"/slfC$ 01 ecs. Computi ng fa c;i1itiE'S m ake it possib le to establish t he chllraet",ris liCi o f stu d y-st a t e OPl'rai ion Ill> II pe- ueul er ease of t ra ns ien t operation . Th us it is posalhle to proc:eed wt t h dy na mi cs no t fro m th e eq u ivalellt etreun. t .e . not fro m t he pllrlir ula r t o t.he ge ner a l. bu t from t he ge ner,,1 equations o f c ne rgy eo nve rsfc n t o sta t icl'l. T he m od el s being se t up in thi s manner- hav e br ough t Ic r t h th e pr ob lem s for s imp Ufr i nU- the rullth p.mntica l Ana lys is of onorg)' conversf o n a nd m ade il necess ary 10 In tr od uce th o notion of specific p ower. It proves i de ntlcnl t h ro ng ho ut the air ga p in p rope rl y d esi c ne d eleelric ma ch i nes lind cnn pro ba b ly se r ve as II ba s ic qu a nti ty in desig ning ECs. T hE' p ee ble ms i!,,'ol " ed i n t he c p t lml ea tlon o f EC5 genera II)' ha ve seve r/ll so lu t io ll8. Tht end pursued by opttmLz ation is to uek tM best solution among man ll pcl~n ttallll poulN~ wlu.tlonJ. A n IlDllmbia;uous probl em doea not n~ o p ti mi zM io n . Optiln izat ion rt> li es o n m a ny met hods , from rn t he r ( oDlplit n lcd a na lytic lind numeriral llp proll chel t o t hose be sed (II I Illm d ee m p iu aUOIl. Rega rdi ng It s m athe m attcnl s t e te me a t , the c p t .i m iaa t.in n probrem red uces In th e pro blem of non l inea r progrllmmi ng . I ts Iorm u lnt ion CIl II be flS fo ll ows : find t he e x trem um (the ma xtmum or m in im um) of 1\ uennne run eucn F ix) dependent 01 1 va r ta bles X" x .' .. .. X n on co nd iti on t h al the t wo k inds of cons t ra ints a re sa t isfi ed: CI ~ z / :S;; H I. !,.., 1,2 , • . . , /l. U, met hod , Ih o per missible lIma of vm-iatlo n or each or l lll' facto rs is hrokoll down i nl') In I'IH:sIll'.'1 w it h n dcf fnite s te p 1I0d n n op t imum \·('.l'Sio n i.~ SO ull: hl by II'lIci ng Iho gr id poi uts s te p l,r s tep. T h6 me tho d , lwill:;:: \' irlllully ~ i m p lll , In-comes rat her clU !l borso m ~ "·III! nu tncrceee in tho nu mber of des i1;:"1l Vll r i;lblos lind {I dCt'l'('lIs0 In t ile slo p lengLh. T ho t ol nl num bt' r 01 LlIC sclntloua I,), 11,ls met hod is eq ual tc a pPl'o)(i1na ll,I)' thc pro duct of the ste ps mllcle for /III \'/lr illblu. t.e . III) X mot . , , tI'I" . Th o met hed co nsumes" fll ir l ~' {:roal dellI of t ime . 'I'IUl- next melhod 111.:lt ha s received wide l\ppl kn li ou ts t he GIlIlSSSc idl'l re la xl\li on (succl'SS ivl'-d isp l n c~ tne n lS) 10..11101"1. T ile idea of l ire method lt es in t ile seq uential S('u d l for til l' pa r ti nl ('~I r,.. m u m of tloo Olltp ut tueeucn for {'a d , va elable of t he sy ste m . l 'lIe l'Carr h is t erm inntcd .....hen t he 'I\;'x\ point of the s pace tu rns out to 00 the er . tr cmum in 1111 (oo r4 in/lle di 1'(>{'1 ionl!, Th o met hod is effic ie nt for IIpp licl1 t,ion 10 Ihe objects in wh ich tl,e (Om l. l io n betwee n t udept'odcl.t \' rori /l bl e~ UOl.'$ no l c~ L"'t . T IIi' sil~ l e lr:t\'er:llt' throll~b . 11 " lIrillhles ca n t hen lend t o t ~ sohu ion of t ho posed prob lcln . •' or t l'o objarts ~'i t h II l orge num ber of ,' a r i/lhiM whirh I' orrelll l(' wit h ea ch other , the met hod peev es Ince uv aut eut , T ile Ine lhod of t he s tee pest s lo pe or stecpesi. descent (hco bi's nlulh od ) ilf a versio n 01 th e grad ien t method . The d irl'Ct io n of senrd . lM.' re i:< oppos ite 10 tim di~tio n of th e grad ie 'lt . If / (.l"n"'l" ) < / Ix,,), t lil' nll nimh al ,on procedure rcuows: e ach ~ I C P is t ake n i n th e SOllla

213

direr t iQIJ unu! f (;>'·. + 1) t ur ns out. 10 be geee tce t hll 11 I (x~ ) a t :Il'~r l:lin s tep. Afttor rc gre!';!iOD 10 U..e potn t r . a nd calcu la tio n of the ~rll,jiel\ t ' PeW. UI(> search ill con ti nued M before. T h is method is m nr o r.Hidell t tha n t he gnldit' n t mgolla.li ty co nd iti ons defi ne on l y t he d ual s pace w hose d imensio n ts eq ua l to n - til . Exp ression (I2 .Hj) then t akes th e form

'is terme d

+

+

L! (6) =- (

n(C,/6 )OI1Ilm\l{ ,\ ~Il

I_ I

1

(12.2 0)

w h ich is Q du al fu nctio n . and bl' 6 t • . . .. 6~ a re dual variables. Here t wo k inds of Ii lleal' co nerr arn te a ppear i n t he Ir mn of in equel it les and ill th e form of equa li t ies. T he fi rst co ns tre rnt imposed O il th e vect or 6 is th e co nd ition of ,f\o l\ne~a L i vi t y of d ua l varia hies: 6, ;;.. O. 6 2 ;;;, 0.. '. s, 0 (12.2t )

>-

T he coost ratnt (12.21 ) impl ies t ha t none 01 t he components 01 -Ii cn n 00 »ega nve . Accor diug lo t he seco nd kind of ltnenr eo ns team ta, t oo sum of c om pon c uta co rres po nding t o t he pri mnl fu nc t ion is e qu al t o u nity ·('t he unr rnalit.y co nd iti on). Th er e is II th ird k ind of constr aints , na mely , vector orthogonalit y .eo ns tr nrn ts. T he)' ap pear as" resuft o f a pplica t ion o f t he in eq ualit y 11.0 which th ere must cor res po nd the sca la r produ ct of e - dim eneic nal

,,, vec tors. T he o rt hogo naHl-y eeedt uo n underlies t he ~ Ileoriell 1)( dua lily a nd req uires t ha t the vector 6 ~lo ng to t ho s pace ofLhogo ua l w il h respect, t o the colu mn vecrora ul til£> exponent ma tri x . Til e assu m ption is Lhll l t he column vectees of t he exponent ma t ri .:t are li nea rl y Ia dc pendeu r. For geom ct rt e programming pro blems wit h th e aero deg ree of d iffie uhy t he du nl regio n co ntliin s one point . T ho soIut.ion of t he d na! a nd the prim al pro ble m re duces to th e !Ml ution of th e sys tem of Hoellr eq ua ti ous , IIIwhich t he de te rm tne nt is diffe rent from ' .0 (0 . Th is sys te m or equa t ions has a unique solutio n. A ~ II ItoSiti ve o('gl'(\o of di ffi c u lt)' , t he rlllitl r ej;i Ull ha s a h ig h degree of nrbu rert ncss . A unrquc aetuuon of t110 sys tem of Iincnr equat.lnns docs not t hen ex ist. Th e du a l Iunct tc n ill s imp ler th l ll the prim ",1

Iuneuon. A du a l prot:Tam o t reee com pu ti ng a d" a ntat:l!s beca use t he co ns t ra ints a re li nea r; the co nstrai nl!l of a pr imlll program a rt' nonlinea r. Th a t is why th e fi rs t stage of the sol utio n to geome t ric prog ra mming prob1@llls Involves th e esn me t re n of t he du a l veet cr sa tisf~' i r)g tho rnli lli ti o ns of or l hogo nali t)' ODd oo rmali ty_ T he sol ut ion of the syste m Clf Ii llClIr equations

61 + 6s +_ .·+6,, =1

" U61 + o u6s

0",,61 + umt6,

+ , . + 0'1,.6"

= 0

+ . . . + 0'",,,6,. =

(12, 22)

0

f ull y de Hne! the dt reeucn of t ho du al vector . The du a l vect e r 6' (6;, 6~ , . . .. b;.) "Offers t ile possi bility of ee te emi nh1i' the dual func l ion fro m the ex press ion

v (6') .... (CI /6:t ': (C, '6; )ei ... (C" '6;,)6~ , \~a

(' 2 2.1)

By t he dualilY th eo r y, masi miz ing dual vJtria bles 6; )'iel d min im· b i ll i i nde pen den t varin ules of t he c bjee t tve fu nctio n. ti_ T he refore . Cl t~I< ' :Io

.. . t:. = 6jl' (6'),

i = I. 2, .... n

( 12.24)

Tak ing t he logs of (12. 24) gi ves t h,c s rste .n of n linear eq ua t io ns i n in dependent vllrin bles t,; t he p r ima l {unct ion . T lte sol ut io ll of t beec cqllnLions dct ur m tnos th e so lt ~ l, t- ror up t.imnl vnlue o r t he c bjeeuv e hillel ion . I n a nology to Hooar program min g , the techniq ue o{ geome t ric proa ra mm ing reli('lI o n t he du a li t y t heory. Bu t ui eee is a dl lteren cc, nam el y, in goini fro m t he pri ma llo t he dua l program , t he nonlinear funct ion (12.6) becom es l inear due te th e tra nsition from t he ngio n o f inde pende nt var iAbl('s to th e regi ~o of exponents. The lh('O r y of geomet r ie program ming is!Ot forth to dcscrtbe eo nvex funct io ns de fined 011 n convex se t. Aecord i ni to ti le ba sic t heorem

or

220

of (' O Il VC.:I: pr ognll ll mill g , en y poin t of II lucrll minimu m of the Iunction is a lso ure point of II globlll m in imum of th o ~iv o H Junctlon. T her efore , t here is 110 noed t o obtatn local extrema Am! com pa re them \.(1 mak e rue ebetec of t he !l'lolJal selut.io n . Tho l,:, si c cr ite r ion of tho me t hod of geome t r ic progr am mi ng is tilt' cbject.lve cons traint . T he objective fllllCli oll , or opt im a lfty erttorin n . is su bject to vario us co us tratnts (ill thc form of equulities lind tnequelttics] im posed o n , sny , t ho cost , rlfm e naicns , a nd overlimIt in dt)s ign ing e lec tr ic rnnchines. I II the tex t above (see &OC, 12.1) a Jew co ust ro m t mathods ha ve bee u brilugh t out . nllllle ir . t he me th ods of hound a ry VA lu es . pe nalt y funcj.lous. und La gr/lllg iaJi ffiu1Li pli o,'s. III geomet ric p 0); f , •. . , t", nrc i ndepende nt. vartnblcs or [he s yste m; and XI ' %" ' . %", ar c objlw tive oxp oncnts . II nv iilg q Iunc tinna l constraints, o ne hal! 10 deflue "I' At' , , '. A~ " .. Aq Lnqra ngia u mutt tpl lc- s . E ach Ln grnngtnn multip lil'l' C(,r responds to t he const ra int woil!'lil. T I' e oh jcct tvc co ns t ra fut cumhines a ll colls trui nt.s. so on c La gr uug ia n nll tl t ip li l'l' "'n Mtled An obiutt~ ~ multiplier cor responds 10 t h is cnust raint.:

ll'

(12,26 )

Where (lo t) reglou of sem'l:h for tuc o bjec li\'ll fu nc UolI ts e pofnt, the sea rch for t ho functi on ex t re mu m nnd cnrrespo nding coord in a tes co mes t o II wclt-Iuunded si ng le eom p m a uo n procedure . TI ".. ohject.lvc ennstmint \:1I.1l1l0t n ppl y 10 nlI prnblems. l t ca n u\:t all II useful 1001 in so l v ing problems or th e zero ,lel!l'l'IJ of (l im · culty. 1lI1d 11 1.~O in expc r im enta l Ile.'1ign and goOl1lfJll'ir. pr ogr Ulllln in~ .

12.3. Design of Electric Machines by Geomeirlc Programm ing The ck$tgn of a lt electri c: mndune by gCQm,etr ic; programming begin" with calculaliofl$ based on lhe ex per im ental: de.~ifPl techn ique, The tntti lll stage comes 10 working fllIt th e program by use of th e known desIgn formufos to /($/ th e fu ncti on in tne region oj interest an d dl'_l'iv l.' t he f irs L system of Huellr eqcaucne covc r i ll l{ lilt II pri or i mfurmn tln n

12.3.0.,0;9 " 01 EI., d,;c MClchino o b y Goome l,k P'09'Clmm;ng

on t he mcchtne. Such a system of uquat tons rnllY tak e cus 'll 1'j

= =

Cux , Cn ""

+ C..x . + .. + Cll~x .. + Ct~ : + .. + C 2 " ,x

",

u.e

221

form (1:!.27)

+ ..

It, = C e,x , -i- C" x . + C cmxm where C-QS 1'. e fficio nc)' '1, an d C n ITt

"',"1

"l,~ i

C,,,,,,,

I

I

""ltl

1 I

'

m,lIf

+ ~ C~ (2j)1

r;

...,lIi

rr (12.33)

';

T h ~ so lu tio n of Bqs. (12. 27) appl ies to t he 5OColid system 01 linen eq ua t ions whi ch describe t ho link between the e lemen t.:s of tho upcnent Ulatrix of the desi gn and t ho pos it ive com pone nts of the dua l vecto r Glib, a , t6~ al.6.. - 0 a l ,,6.. ." 0 112.34) Gull, Gn 6.

+

+

n...,61

+

+

+

+

+ 11",.6 2 + , .. + a"'ft6"

""" 0

where 0 Il' •. . • 0 ",. tire t he ex po nent s of i ndep endent "Bria hl l.'lI in t he opt imum desigll j lilt 61 , . • • , 6. lite t he Jl(Is ilive com po nen t! of t he mi n im iz ing du e l vect or. T he lIys l.(' m 01 equ a ti o ns (12.34) rep rese nts t he rela ti o n be tw een th l' expone u t, ma t ri x of t he des ign a nd t he posi tive com pone nre of t],tI m ini m i[i ng du a l ve ct or . Th e last ecl umn ve cto r of t ho ' ex ponents of t he Inde pe nde n t varia bles of t he des ign:

+ a . , z. + . . . + a" "'::,,, = 0, + anz. + + a"'tZ", = 0, a l~zl + o."z. + + a",~ lI:", = b", 01lZ:!

a u z,

(12. 35)

where .::" z" . . .• Z ItI ar e th e logs of i. hu abso lu to va l ues of independen t var ia bles; 01' bt • • • • • OM e re t he logs of t he p rod uct of t he posi t ivo compone nts OJf t he mi ni mj ~ing dua l vector lind the object ive fu ncticn minus t he logs of t he cocrnc re nts a ll t he va rln bios or t he posn.i vc compon ents the objective fun ct io n, T he des ign process yie lds fi llll i resu lts by t aki ng the a ll! ilogs of t he results of th(' so lut ion t o the third sys te m of linear a lgebra ic eqn e t tc ns . As An ex ample , cc ns rdcr t he prob lem of sea rch for the geo metric dime nsio ns of a n energy convertor t hnt ensure t he s peci fied si e uc and d yn ami c out put ch aracto nstt es a nd effec·t It maximu m s/ld ng in copper, 'The objec tiv e fu nct io n is g ive n bi' (12.30) for d et ur min ing 1110 mnss of th e wi nd in g. T h is funct ion dep en ds on f Ollr var iables, D/, l, b , . and h n. Deno te the expo nents by X" x~, %• • a nd I t for t he va ria bles b•• h n, I, And D 1 respe ct ive ly. T he ex press lo u for cons tra in t func t ions will t hen taka t he for m of (12.27). The snlution of (12.27) enables us t o for m UIC o bjective censtrnint

or

C / ( o: 'h~;I"·.m'} ~ l

(12.36)-

The seco nd eq uatio n in (12 .34) will t he n t ake t he for m 16, ~ 16. 1- 16. 16 t ~ 065 ~ 1 161 -t- 16. 16. -i- 16t - X16 1 = 0

+

+

11\

+ 16

2

-i- 16.

16,

-t- 06.

3

0&,;

+ 26

t -

+ 06 -t- Ollt + Dill + 16a ~ 06. -

0 X;/>, = 0 xt 6. = 0 Zt65 =

(12.37)

Soh' i ng (1,2.37) wi t h co nsideratio n .Iur (12. 27) g ives an op timum. (m inim um) value of th o ob jectiv e funct ion

Geu mi n co (

kC6, "" , )' . (ZkC~I:., 8 )'. 6.

X (nkeuk"t!{,

6J

)'. (:t~CU3l.

D, =e"

( t2 .4.3 )

T he m('thod of geome t r ic p rog n"m mi ng ca ll ~·iclu the r(')llt ions 1'>401 "1*'n t he c ut put lind in pu t d llLra c tor is lj(jIl of lilt' o bject u nder s l lld r in t he form fit th e sys te ms of Ii nt'ar a lgt'-braic equa ti ons . Thu IInnl )"si:5 or b lll.'.rg~· co nver ters w!tll the a id (If th e (';\" Il()II\' I1 IS or illll(' Plw d t' 1I1 vn rie blc s is the ~ llI!rn li znt lol\ o f II higll6t order I I,,,,, ill tho case for co nve nt iona l medcls . '{'l it' o bjec uvc exponents of ind epcn dOllt va rtah les in lil a ohjeclivo OOIl;llrll.int a nd the poetuve CO lll pol\('nlS or t he mini m izing dun! vector fpc clcetr!c mnr l' incs of t he ."l1 I\l(, se etes cha nge e ver ins ig n ificlln t in ter val s . T h is pe rm ils l'- ;\" !on d lng lh o dnl n .1/1 a t horou ghl y clcsjzllcd machine mode l 10

\ 3.1. Evo lu tio n of Syllem; o f Automatl!d Oesi g n

225

o the r dc~ig n vers tc ns or ct cct rtc machlnes of th e sa me se r ies, th ere by cultillg dc w u hoav ily t he ti me requi ted for t he comp uta tions. T Ile des ig n te ch n ique ba sed on geo me t r ic progr'a uuning follo ws fro m the geneml lae t to n of the theor-y of mathema t ica l s imilit ude, where tho desi gn procedure relies on tim element ar y laws of li nea r alge brn .

T he me t hods of gec me t ctc p ro~r:\ m rni ng t oge ther wit h ex puelm c nual ues ign me t ho ds hold mil ch prom ise for the so l u tion of op t imizatio n pro ble ms . T hese me th od s do not ce:rta ili ly ru le out t he np plica tic ns of othe r cpti mfaa t.lon me tho ds .

Chapte r

13

Automated Design of Electric Machines 13. f. General Points on the Evolution of the Systems ot Automated Design I n th e USS I{ t ho use of compu ters for d es ig n of e lec t r ic ma chines was begun in the lnte 1050,;. In tho mid- H170s a ten dency evolved t o illtogrRte sepa rate. desi g n ap proaches a nd rorm tl s ing le me thodolog y t o promo te opum um de si gn of etccu-re mac hines us iug the concept of (I ~ Jleril 1iwd enor!:~' con ve r te r . H owe ver. t he general d es ign proeedure li t the t ime llU'gely re mained manua l desp ite the fAeL t hat computers held 1\ defi n i w ple ce lIm!lllg o l her desi gn mea na. In the modern peri od of lIppliCjltlOll of computing fflciJi t ies t o the processes of des ign of cteet- te rnnchi ncs , the trend is to rearra nge fu Hy t he d('sign techniq ues on t ile bas is of automated desig n sys tems so as to tra nsfer the dMiqn prob lem i n t he for m of II rnuthe mauce l model on to a co mpu te r to ·ge nernt e so luti o ns. T hus , princi pall)" new desi g n a pproaches h llvo come i nto being. Ob v iousl y t high-qnnli ty des ign work mu st be done in Lh0 shortes t t ime possi ble . c r berwrso t he ide;l,' pot in to t he project and th e e ngl net'Ci llg sotc t tcns will be o bso lescen t. even before t he ma ch ines como into service. A de.:lig n insl\[fi c,ie ntl y wor ked out a t t ho ellrl y stagea of its developm e nt en t a ils 11 ]c Dg t hy per iod of " upll:flllii ng" t ho pro to t ypes or ev en re mode li ng t hem at. t he produ cti on stage , t here by add ing to t he ex pe nses lind ~rotrnc t i ng t he d O$ig n impl ementat ion . A pr- tucip le o bst acle t o th e i mpro veme nt of the q ua lity of des ig ns an d red uct ion tu tho ti me of the ir d evolo pme nt is the inco ns istency betw een com plex moder n mach iner y a nd the ol d me t hod s an d IUea nS 1 ~ _ O I1l8

"6

of tles ijt:nillg'. l\t lh e nge or le clm ica l progress. 11I1 Increased comp lu it)' 01 des ig ned objocLs is Inev lta b le, ",nd one (:Ol'Wlillly ca onol TIl! ! t he q uu lily 01 d ~i C D work .. nd e ecclc eatc t ile dQSit:n peoeess Ly just incre asi ng t he number of dClli¥1I offices. Th o problem is a mell. ble W the so lu tio n o illy with t he .id of mRlhe mfl ti u l lllelhoos amI eompu1IOll f~ cili ties insta lled at project and dc.sil:lI servj ces , milnu{ael Uf Ulg Ofio Diu tio ns , a nd at var lces plan lS. Th \! most eff octi ve des ign guideline is 1.0 con ve rt lrom e u tomaucn of indi v id ual desigllS to inlegrll tcd illlt(imal io n evo lv i ng f (lf tile p urpose a utoma ted deefan ! )"s tllms (A DS.~) . The AD S is lin o perat ionn l eng ineering s ystem s csocte tcd w ith deJIign c tnce su bu nits amI inten ded to fu lfi ll the nss ig nmcnts by t he availa ble e uto maue mea ns wh ich for m II s pec ific (', " lllp l" x . Tho enmplax e nsures me th odteal . program m ing , OI nl ori nl , i nlnrm ation, fw d orgd up on 1,0 me ke a e hctce o f 11K" op timum des ign cut o r II. variety 01 t ilt" d eaig n ve rsions,

13 ,1. E~of"ti on o f Syste m. of Au lo moh'd 0 0. 19n

229

e ffec t, o pt imulIl unifica t.inu n nd s t und a rd tza t.ion of s ubassembl tca a nd pe rt a of t he mach ine . T h is crrcum stn nce considera b ly ca mpI tcot ea th e proble m of dlls ign work. T he re ason is t h at, wh ile conv entiona l problems gc ncrlllly hnv e n clear-cu t mllLhe nl/lCic,nl s ta teme nt ilnd Me solv a bl e by form a l me tho ds , lh ~ proble m of search des ig ni ng shows u no nfor rnai cha r acter and , .hence , is not a mena ble to the solu t ion by etrecuve nurn or tca l mll-l/lo(1S. Nevart.hefess. \1\'(' 11 ill the abse nce of Ilonfom ln l ma t hcm at.lcel mot ho.ts , ure p l"ag mnt ic basis for t he solu t ion of des ign pr oblems ean be man- mechfne co nverse t toun l prn ccdurus . T it/.' d inlog is ma de poss ib le b)' t he method of search fur nlte rn anves . A s he fo llowA t he course 'lf t ho so l ut ion to t he probtern, the d~ ig Jlc r (',1111 I,nke n numb er of posafhle dectsion s Ilt urc g ivon H Il/N of the solunon. Th e desig ner m akes :) cho tec na Ito adjusts t he sv st em with t he grap h ic termina l for t he st ep- by -s te p snlut.ion . T tm eflcotIveness And f1o)(ib ility of the s ys tem de pen d 0 (1 whet he r ttl" net th e sequence of nc lio'lS suggest ed t o the designer ca n affor d H s uffic ie nl ly large number of possi bl e solu tio ns II"t ea ch step . I II t h is eonnect ton, infor ma t ion sup por t of t he evercm Acqu ire s mu ch Im port ance. One fin ds it ndv autegeo us to Accompli sh inform ntion a id s as n se t of prugraurm nble m odels whic h de scr tbe at an de ed desig n elements lind sim p lc geolllclric. fi gures an d also aa sem antic mode ls reflect in g the hiera rch y fl lI n s truct ure of cleme nts, par ts , a nd ll11 it.s of, t he machine being" dt·s iglll.'d. Suc h II roprcso utnt.lon of g rup hlc nnd somant ic info rmnt.ln n hn s it s rcote i ll tho preli llliullr y n nnlys is of lh e geome t.r-ir, for ms of t he pnss tb le des igns of a mac hine find us el emen ts, a nd nls o in t he syst emat.ic 'a naly sis of th o ma chine dcsig.\ l ll ~' u u t lind h ternrchy of t he Isola te d el emen ts ( ill the for m of n gra ph of poss ible solutlcns). Ap R!"t fro m the program s 01 t he geo mot ric forms of ele me nts end par ts , th e grap h ic inform nl ion aids. of the syste m nlso i ncl ude the progr ams of the pr oj ec t ions. Gross soct.ions, re pr esentat.ivo dtmcnsions a nd di me nsional t olera nces , ra qut re ma nts t or winding sur face clellllli " ess . s t a nd ard t ex t, informnt.io n. e tc . T Ile AD S of etecu-rc machi nes (AD S E M) ha s n geue rn l pr ogra m of drnwlng . It e uvtsages work on vnrinus pr ohl ems lind collects a nd stores i nforma t ion . Th o prchl ems of et orsges, search, nud proooss ing of tla la acq uire pr imary im por t an ce j n. mode r n desig n sys tems nnd ha ve a substnnt.ie l effed Oil t he str-uct ure nnd (he pri nci ples of acti on of th e s ys te m as n wuo lo . The fil es of data nnd progra ms direct ly int e nded to peovjde for cent.rellzed st cm ge und scorch of Inrcrmauo n nnd al so to estlllllish li nk s w it h a pplica tio n program s wldcl~ pr ocess this in form at ion form a bank 01 data . l n systems using data ha nk s , up phcat.ron progr a ms rece ive da ta lor pro cessing not from external du t n cerriers uut Irom ro m rol syst ems of the deta base.

'30

Ch. 13. Avt om eled

.o••lg n 01 Eledrle

M""l\jne .

One of t ile im por ta nt- problems of ADS EM ill t o create" t he da ta bank of elecrromechnntca l sot uuons. For t h is i t is necessary th at deaign an d de velopmenf work carried 0 11 in t ilt' rte ld of en ergy COn ver ters sho u ld comply with tho requirem ent s of tho ADS. The proce dur e in t he ds valo pme nt or II new energy co nverter should fo llow th e ado pted gl. ideli nes on\' isngi ng t ho st orage of finlll reeults (II th e- dat il bonk for fur ther uso. T he bank ca n store infor mat ion i n li braries o r~ n i 1.ed in a n hiera rch ie orde r accor di ng t o t he ty pos of EC or (or each leading resea rch insti tute (F ig. 13. 1). T he lib rary ca n con t ai n dat il 011 deslgn Fig_ 13.1. The general ~ehem8 tl c of a of Ee ele me nts a nd s ta ndard data bank for energy cOll vert.er~ so lutio ns. Th e evolution of lin ADS EM is Il lengthy process a nd call s for fur t her deve lo pment in the field of electromechamcs and mom coor dine tc d work nf deSign dep a rtm ents at plants a nd research i n! Ilor t io05. t he cho ice of fa!~c nc r.!l . etc .. n ru l »tsc i n th e eo ursc of briJlp: ing 1110 d es j ~ll to tile Il na l Jenn III com plia nce wit h t he fl'q ulI'eme nt s of the naelg nment . Afl('f co mpleLion of t he gcnc:ra l d ra win g of th e mecn tne , t he seco nd stoge o f des ign t a ll follow , wh ich invo lYf's dC'vr lopin:.: the eesc m bly u ni lS arul pArI.! of t he co nlll.rllt tion a nll de le rnl il1i' lg t h('!r maas, d imclllliolls. lind to le ra nces req uired III tho preduc rto n slagI.'. Thi s dooc , the a utoplorter ca n fi na l!y ma ke up t he work ing draw i ngs i n co mp lilUlcc wuh the roq uirem(!nts of sillo ndllrdi:r.alion . T ho s u bs}'. . t ern of d rll\\'lIlg fllc ili t ics pro v id es d es igll lind prod uct ion d ra wings a nd s ubs ta n tia lly cut! down th e lead t imes. Tho deve lo pment of auLoma l.(od lJe:li{!1l l!I ~'s lf' J1HI poses a num ber of co m plex pro ble ms . Of m uch l!i(:'n if iu nctl is th o es tn bfisiunent of i nt er na t.lo ua l p roKrllm libra r ies whi ch would pu ll in ~iC ll L jfk peIen tia l 01 e n" " Il.>e rl!' for t he so l ut io n of t he most im po r t a nt pr ob lems o f alecr rc meeha ntcs .

13.3. Hardware of Autom ated Design Systems The ctnlra l p rO-]In. Utc1l·

lion or l't"O,d. ; t.;G _ 1I I

25

-

,x

:le

. ."

.,.

.•

.,

•• ~

' flO 50 2U

' 00 50

teo

0 .1&9 HL 2

1I .1&!J 14 .r.

(J. Hi\)

o I [,!J

fU 5(1

O. t S!!

11 .2

14. 2

1-1 .2

1;-1.4

<

•

t n> ~

~

~

so

"

eo

28

,•a

,-•...

<

<

••

.,:Q

<

-e

•

1((1

'00

HiJ

O.WJ 101.1

o.isc

.v

100 50

",

."

~

.,

so

so

50

es

",

15.4 Tdbl.. AIl.S

Gainli of Amp Uflers of the Comp lrtet Analog

1r1 =k l Irs = k.

0.5 u.55

0.5 088

0.5 0. 67

0 .5

c...js.j

0 .477 0 .7S 0 .6 0 . 714 o.en 0 .53 I). S!1.1 f). 57!! O.I,M

0.43

~~ = k.l ~

I

I

~1 ~ - ~I ~

7 . t 2G 9.8

~ n -~ l '

fl.!)?

~·a .... ~,

~': = ~IO

~3 =~ 1l

k ,, - k 1t

:U

k u ""' kr,o

LSt 2.15

~, ~ = kzo -~' ~ s "": ~-H

, ,

6.4

3. t4

4.23

,

1. 31 1. 56

1.t 9

0 ."" 0 ,86

4 .07 0 .421 2 .41, O . ~ )()

:{ . t !,

3 . t ' s igo n k}-l ",.. .:, X 14.65M X 50 = 0 .293

I II cl fl r Hyi ug- pllilll 1 of '·ho llrn b l ~m. we sho uld lir e 10 it , ~o ll s-i ,Il'T_ i ng the specrl red varin bLo par cme t urs , what coelricicll t-s in the sysrem of equa t.lnns rHO vnrtable , the n rucalc.ulutc th o cOl'rellpo ud illl; gain recrors nnd s umma r-ize the cto;lu]ls in Ilw tn ble . _,, ~ 1' " r-xamplo, T nbfe A IV.::I g i v e~ t he ros uns or Ihl1 ce lcu lutlu n of ~u i1J fatl ol''! h01l 1l1 be milde u p for t ho llss ig nml' nt vlll'innl iuv elviug l.JH~ sol ut io n o f ~h l:l mo t or equat lous e xp ress ed in tenas of curre nts . It IS now necess a ry to a nalyze We ob tnined design nmtrr x lind det cemine t he expedie nt seque nce o( it s iOlph.• men t a l.Inn on a ny nnnl og comput er SODS to reca lculate u sma l lae num ber of t ho ga i n fact ors in go ing from on ", t r ial 10 t he o tJ,er , T he t r iol i n expe r imenta l des ign me a ns t ile calc u lat.io n on a n a nalog computer of th e ebsrecr ertsucs o f the tra nsien t and slea dy~stll te pro ceseee ill t he i nduction motor at t l\e fi xed vnl uos of its parnmetor a. T ill" u s e d va hles of the motor parameters are ch ose n for eac h assignmon t. v nriant from T u ble AI V.2. Since in t ho co m puter all a los;: tho ampfifieea s how zero d r ift. t ho exper iment m ns r he re ru n 11 1· leas t th ree tun es and th e results entered, in t he tab le. 2. The e xper-i me nta l des ign e nables us t o obta in a sim ple rclall o n betwe e n t he va r iab le pa rame ters of t he m acbjnc and u s cbe rac toeistics. T h is re la tion has the form of a po lynomial

!I "" bo+~ bl2'l + I_ I

f

I a' {II} . t he d l!scr ip t io n is cons idered Ina deq ua te 10 t he o bj ect u nder analysis . T he chec k of t he hy po thes is for adequa cy is ca rr ied o ut usi ng F isher ' s v ar iance rat io tes t . T he F- Iest perm lU chec k i ng the Dull h )'pulhe.sis for the eq ua li ty of tw o IlC ne rali zed Vll.ri~ nce!l an d gt {y } i n th e c.nse w he re 5aDlpli ng ",'or ia ntes S: d > S: {II} = st. T ho F-lellt is defi ned as th e va r ia nce rat io

S:.

F = S:"IS'

H t he cefcule ted va l ue o f t he F- te st is s mAller t ha n its c ritlce l va lue fou nd from T a bl e A IV .5 f (}l' the cor respo nd ing deg rees of Ir eedorn , " a d = " Qd - N - d nn d "tool = " s = N (In - 1). at Hie s pee tfied s ig n ifica n t lev el qnd 1% ). lh e null hyp ot hesi s is accepted. Ot hcrw ise t he h y poth es is is rejected an d t he d escripti on Is eo natdc rod lnadequata t o t ilt! o bje ct unde r st" d ~' . If t he hypolhllSis of ad eq ullC)' is rlljec ted , Il sm alle r s to p of v e r tauc n is til ke n lind t h\:l e xperime nt i.:! ru n a new. :~. O pti mum param e ters o f t he mo lor I\r8 fou nd us ing the pol ynom h Ils defined in t he tex t. of point 2 of lhe pro ble m . Tho opti m i2 ~ ­ tio n c rit er ion an d co nst.NIinl run cuc ns lire c hose n for eac h asslgnInenl vuia ut. from T able A IY.6. The po lyn om ial s for lJ lnd cos « in l\lIll.!Sigmne n t variants llN! t ho s s me and equa l to

= 0.8t5 - O.OIV - 0.0221"" fJl = 0.83 + 0.036"" - 0.018r' + O.026r' r'

lJ

cos

I n the typ ica l dea lgn proce du re. the c p tlm um p:lrllma lers are d et er m tued by t he se m igra ph ica l ml'tll od in wh ich tw o pa rame t ers are va ned withi n the limit s set up in t he ass lgn me nt . Consi de r the WI )" of es l.imoLi ng t h o op ti mu m parameters . ASIIl1nle that t he des ign e d ex por tmon t co nd uc ted for a cc et a!n t yp e of i nd uc t io n motor ha s g ive n lh o poly nomi nl re ln ttc us (or the Im pact current . im pact lorqu e , a nd sta r ti ng (s peedu p) ume 1lS Iu nctfo ns of t wo peeama ters. T hcStl re la t te ns nre of th e rcem

I ,... = 4 .5 - 0.31"" - OAr' + 0.1r''''= 3.5 + 0.5"'- - 0.5r' '.1 = 150 - 12,... + 2r' + 2r"r' ] ( to-I)

M,.

where r' lind r art' th e s ta to r lind rotor wind in s res ist a nces , respec th·o) y, ex pre ssed in ret euve u n us . Wh llt we ueed to det e rurinc are the va lues of ,... an d r' I t wh ich the s tM ti ng time is 1'1 mi n imu m , with Ih e oonstrlli nts tM- ing imposed Oil a nd M I•. 10 Its mAthematica l form , t he o ptim iza tio n prob-

I,,,,

I I -Oli n

,,, le m ea n bo wr it ten

;a:;1

'. , =

50 - 121"'

+ 1r + 2r' r' ...... mi n +

, , ... ..... 4 .!i _ 0 .31'" - O.4r' O. t r"r' + O.'>r'" _ O.Y ...... "

~ 4.~,

l,/ ,.. _ :S .!> H _ 1 "i; ,.. 0;;;;

I~ ' I

+

..

"

1 li n d _ M

....

i " ' ,.....,; .... t {t il. Yl'lr l " Uon r ang t tl ll ~

up t h e ! 1,cl or;,, 1 " "-lJ'Cr i·

m c n l .''l' hfl SP"C\' ClluHnod wi t!lin

t ho sq ua re is t he inirinl [ichl of t ho perm tss tbte ~ (/l ll ~iO rt>'! w i th t he cn nsl.rll i ll t.! d isf('!S'lI r dl'd .

Till.'

N

ri ri< ~

f ,,.. .... 4.5 -

co ns t rai nt 0.3 r: - 0 .4 r'

+O. Ir'"r" = 4.5 represents an tu toeeept of th e hyperbola brnncll .lf LV which p ll S!l',; t h rullj;l, fl (',cIIIrn 1 poin t Il n~1 0 110 of th" veeu ees . This see uon ea n be pl ntl l~1 U)' bri n:;:in~ t ill' equation for in to Cll nonic,,1 Ioem or by .!i u lJ.~ t il ll l ing ( t'rt lli" ( h ell values of rand r . Tho cnnslrni nl -l1,.. = 3.5 0.5"" - O.!".r' ~ 4 Pig. A12. 1'IM! h\l ..dilJll'U5ionaJ raetc r s~ for ~rillbles r' ulul ,.

I,.

+

represents au intercept of liltl li lle PLQ passi ng through the points with coordi nates (- I; 0) lind (0: ..L.l). 1'110 cons t ra i nts so impOlWd li mi t the Held 'If p~ r m is~i lJ ltl sohnioos Lo n polygon PLNG. T I ll) rel" l io n 1,1 - 1,jO - 12"" -"-- 2r + 2r""J (10- 1 )

defi nes the fnm ily of C.lI rn'!' rl tlJ,. A : H: , A : IJ, . , ., et c. 0 11 ench of these ('lIr\"l'~ III(' \'lI lu0 of t . 1 is cousrnnt , I n rno\·ing from. My. AIB t to AI8~. thl' value of t une t . 1 d~ )"! . T he point L is tile ectulio n Lo 111(' problem s ineii' al t h is poi nt th e um e t .. ta kes 11 minimum value upon !ati!fying the to lt !ill mi nl1l .w l lip (III / I.. I nu .11/.. . Tllll coe edinntcs of the ,"tl'iIl Hlln point or the value s of t he op timu m parameters of lhe Illari d ne ca n 00 found dirl'clly on t ho gr it! of Fi g. A12: r:1'1 ..., _ 0.45 . r.P/ = 0.58, .1 . 1 = 4.16 9

'"

Append ices

4 . T o osti mat e tho accurac y u f t he svl nUoJl, we ~h oll hJ co m pare t he cnlcula ted d ntn with the simu latio n result s for th e u ptim iza tio n crit er ion IUld con stretn t runcucns a t t he opti m u m poin L. F UI' thts it is necessar y to conver t {rom th e rel ati ve u nits used in .les ig lling the ex perfme n t to t ho real uni ts. T rw eo nversl o n is acccmp l tshc . l . hy li lt .' Iormuln

::.;;, =

X i ",) IX j

( XI -

m

~

X j m in

where :1:-; is Ihe f lllln in .!!: v alue of ti ll: mot or pnr nme t nr- in relnti ve uo its: Xi m is I hi' meA n basic vnluo of t he vo rtehle pnrlUncu,r of the motor: ;1'1 q'ln is 11m lowe r li mit of the Vllrillbll.• par a me te r: nod Xj is t he runnhll( value of the mo tor para mete r in rue l unil.s , T h us, for r T vru-ied over t he ra ngl' of IIp t o 20 % (sec 'fa bl e A IV .:5). Xl

= r' =

a.s.

ZI ,Jl.ln

=

,. ;" IJl.

= :'l . .1:1

MGI".. Hram. l t .

':.- "6

j~- I~

I" I

l~ _ j~

3 .5 7

0 . 2711

0 .2li3

0. 2119

a. 7

0.2.85

0. 20

0. 28 11 .28 0 .28 0 .28 0 ,28

(1.285

0 .2G

3 .~

o.285

3 .' 3.' 3. 7 3.5

0 .28$ 0 .285 0 .285

0. 26 0 .2(; 0 .25 0.25

'1. 5

0 .185

3. 7 3. 7

O. 27 ~

3 .7

O .Z&~

0.275

0 .25 0 .2 5 O.2G

o.ac

" 3 2

,

3 3

O.Z8

2 3

0. 28

a

0 .28

z

1).28

3

o.ae

2

".

Appond;cOI Tioble " IV.1 (0 .8:1

M lm ... 0 •8

1_/ 0;;;

~,

1 ,m ", In '1 ",."

c()5 q·~ t'.8

'1> 0 .85

11m ';;; 0. 35

C(' 1I1 of Ihl't>rr of l'lftlric ,,,,,ch ine:5. 9-1'

fndllCt lnn

mkh i ~.

n',

ph msor

loll•. A.

Pqtl,v:alO"f\t

c i~u i l

,j~

d l~l'lIm

l>f, 5G

s., t4

losi f)'ID , A, G. , 13

Itereuve 'lIl'\ hod, 21·28 " I rillb y . M•• !I F.... d~ r ' llllolor. 0·10. 38

J OIIl':!. I. Po, 13

FIeld pqullt lnn 5. \ 9-31 .'i n ;l e d ilf cn:n C\! methOlI (F OMI. :!t\-31 .'i ~h e r· ~ va rillncll . nlio (F-t f"lt). 195.

K~ I) lY ~ Il~k)' ...\

206, 21(1 1'Iow

di ~ l rl b\lH on

In eler.trlc machin e ,

rs r cr te eue. D., 13 I'ollric PIl rlu. 14 Fre'lucnc)' cOIl'·crler.

~3

. £o, 101 Go, \3 i:\~ ~ov$k )', E, Ya., I ~ Kir chhoff's aecollil Jaw , II \ f(~ P I',

KOlI \{'n ko, M.P.. HI KO"llc h, K .. \3 KhnJ$hchev , V. V., 13

KI'Oll . G. , \;1, 35. 83 Co p " "",," ifon llil)', 136 Centro !Or, auto mot ive . 11 eO"w . t in .. I)·Jl(' \'00 de Gral ff. t:l ,

1S3 Indlld o e, 178

rMo!lWloh)'drodsnam Le (MHO), 3 138 . 112· 174. oOll:"inll5O\i dl l 'l'olu.,.ee, 100 GI'Oll>l' l r ic pr"Gu mmin;;. 214·:220 GOl"o!v . A• •\ .. U Gra"unoe. 1.:.. 9 Gnuu1", L. N., 13 G""'rickl. ,'on 0 . • 12 J-1 arm Oll 'e:I. hel('f'lldyne-lre'l ul105O v. )I . v.. 12· 13

Midline. COlIIDlUllllllf. 33 de, 122-125 donble-rulltiO". 100 fdu lhcd , 32 ;Ildue lion . 32 JIlul hw indl nr . 116-111 prl m;t ive. 31· 38 sync hronoU!l . 55-5 1, 117-122

-a

It>d.. wi t h with w ilh

t breoe t1 eg l'('t'S

06' fou r t1 ('gT~l~ fl ve

06' dl'greeli 01 Ir l"E!dom, 06' degrees of free dom , bypot he· " i;>e

wH h

wi t h

of frCCd\>lll, 108 t1 egre e.!l of fT e(!dom,

1\

t ica l. 168 Magn et ic field strengt h, 2 1 Ma gneHc IJUJI d ensit y , dc fin iti on of,

20 Mal)ll elshto m , L. I. , 14 r.lat rh , d('Sign . In illveT5e, 4 8 , 19 1 noise , H ~ of com ple te fn', N. A. , 13

Vari.DU. expe rl meot, 20Ii

Inl'dequa" L'l ica l loHs. Our lul dn'S!I Is : USS R, 12 'J(l20, MQ."1::Ow, 1- 110, C S I' ,

I' " rv >· Hh hsky Pereulck, 2 , Al i l" I' u b l; sh('l'll

Other Books f or Your Library

C YB EJ\ Nt::T ICS IN E LECTnlC POW EH S YSTE :'IS

P rof. V. A. v ent kov (General edito r) 'MI' boo k cov('n r.la ti vcl )' ntlW maU-ri:ll not )'ft full y wo rked ou t ! ei t'lIt Uical l y . kcllnologlea.lly ud melllodo logiulJ)·. l>ll rticul-

arly

,)(I

reg. rd& rOl'l'easti!ll:', pl'lRDinll and contml or la rgE' p..wer

'!l)1' l.t'IIl.ll _ Becauso 01 Its novelt y and some cODt rovt'uiA! vic",lI. the .!IlIbjec l- In,a Uc r lark.!! un iform ity o f p ~n l .t i on , g elK'r3 1 W !M:l' PU Rnd ten lll n()l~y , " hesll li m ita li ons e an onl y hI' rem oved b)" furt M r c:olll'C-livt effm't, and Ihe a ulhur;r wlll IIpprec-ia lll a n)' sU.!igot'l;l ion.!l Irom n·alh:r>j _~tuden t.', lect urers a nd ~'nll' i n~,\..ra,

T his book hllS.been written U I' stlldy a Id lor st udents at power and ek c trieal engi neeri ng eoJl~d lind depart ments , especially whe re t he c..rrleulo includ e ecue es 011 va rlou s aspectll of PQwersystem eybemet tce. Apart rrum ~ ~lId e n l$ , t he book may be c value tn J'{'l!l.'nreh work",rs , p' lllt ~r;uh'lI! e:s and enginl.'Crs euncernctl wit h powc r-~y~ lem con tml nud alJi('d tJe](l., .

A N I N'I' HOD UCT IO N TO COM J' UT EI\S

N . Sergeev , N. Vashk avleh Thi .s book coven! the oire ui tr y an rl opeta llnll pri nciple!! 01a na log a nd d lg ild COl.uputer!!, spectal - purpes e computing dev ices. ma.-hl ue!! end sys tem s. a mple s paceis devo ted Io a basic the ory of s imili t ude and slmulatiO Jl, and a la i rl }' detailed deserip rlcn is givon o f basic Juac t tona l elemerua , assem bhea and un its wh ich make up I)' piea! a nalog an d d igil al romp uteTll. in ro nj unct ion with a n in troduction 10 the mat hl"matieaJ and logicnl basis or elec tron ic .lig itaJ ecm nurees. Se parnll" S/!'Ct'OM deRi wit h biock-d tag mm sy utbesl$ or a nalog computeI'!! to sol ve a lgl"l" roie, trensecodentul , ord inary a nd pa rt ial di fferen t ial (!